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Loyola College M.Sc. Mathematics Nov 2006 Linear Algebra Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034  M.Sc. DEGREE EXAMINATION – MATHEMATICS

AA 18

FIRST SEMESTER – NOV 2006

         MT 1804 – LINEAR ALGEBRA

 

 

Date & Time : 26-10-2006/1.00-4.00           Dept. No.                                                       Max. : 100 Marks

 

 

Answer ALL Questions.

I ) a)    Let T be a linear operator on an n-dimensional vector space V. Then prove that the characteristic and the minimal polynomials for T have the same roots, except for multiplicities.

[OR]

Let W be an invariant subspace for T. Then prove that the characteristic polynomial for the restriction operator divides the characteristic polynomial for T. Also prove that the minimal polynomial for divides the minimal polynomial for T.                                                                                                    (5)

  1. b) State and prove Cayley-Hamilton theorem.

[OR]

Let V be a finite-dimensional vector space V over F and let T be a linear transform on V. Then prove that T is diagonalizable if and only if the minimal polynomial for T has the  form  where    are  distinct  elements of  F.                                                                                    (15)

II )a)    Let V be a finite-dimensional vector space. Let   be the subspaces of V and let  . Then prove the following are equivalent.

  1. i)  are independent.
  2. ii) For each we have  = {0}.

[OR]

Let  be a non-zero vector in V and let  be the T-annihilator of .Then        prove that

  1. i) If the degree ofis k, then the vectors form a   basis for.
  2. ii) If U is the linear operator on induced by T, then the minimal polynomial for U is.                                                               (5)

 

 

 

  1. b) State and prove the primary decomposition theorem.

[OR]

Let T be a linear operator on a finite-dimensional vector space V over F. If T is diagonalizable and if  are the distinct characteristic values of T, then prove that there exist linear operators  on V such that

(i)  T;

(ii)  I=;

(iii);

(iv)

(v) the range of  is the characteristic  space for T associated with

Conversely, if there exist k distinct scalars  and  k  non-zero linear operators  which satisfy conditions (i),(ii) and (iii), then show that T is diagonalizable, are the distinct characteristic values of T, and conditions (iv) and (v) are satisfied .                                                                         (15)

 

III  a) Write a note on the Jordon form.

[OR]

Let T be a linear operator on  which is represented in the standard basis by the matrix. Find the minimal polynomial for T.                                  (5)

  1. b) State and prove cyclic decomposition theorem.

[OR]

State and prove generalized Cayley-Hamilton theorem.                                   (15)

 

 

 

 

 

 

 

 

 

IV  a)   Prove that a form f  is Hermitian if and only if the corresponding linear operator T is self adjoint.

[OR]

If  , then prove that .                                                    (5)

  1. b) i) State and prove Principal Axis Theorem.
  2. ii) Let V be a complex vector space and f a form on V such that fis real for every .Then prove that f is Hermitian.                                       (9+6)

[OR]

Let T be a diagonalizable normal operator  with spectrum S  on a finite-dimensional inner product  space V .Suppose f is a function whose domain contains S. Then prove that  f(T) is a  diagonalizable normal operator  with spectrum f(S) .If U is a unitary map of V onto V’ and   T’=UTU, prove that S is the spectrum of T’ and  f(T)= Uf(T)U .                                                  (15)

V  a)    Find all bilinear forms of  F over F.

[OR]

Let f be a non-degenerate bilinear form on a finite-dimensional vector space V.

Then prove that the set of all linear operators on V which preserve f is a group under the operation of composition.                                                                    (5)

 

  1. Let V be a finite-dimensional vector space V over a field of characteristic zero, and let f be a symmetric bilinear form on V. Then prove that there is an ordered basis for V in which f is represented by a diagonal matrix.

[OR]

Let V be an n-dimensional vector space over a sub field of the complex numbers, and let f be a skew-symmetric bilinear form on V. Then prove that the rank r of f is even, and if r = 2k, then there is an ordered basis for V in which the matrix of f is the direct sum of the (n-r) x (n-r) zero matrix and k copies of the 2×2 matrix

.                                                                                                          (15)

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Loyola College M.Sc. Mathematics April 2008 Linear Algebra Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

XZ 25

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – APRIL 2008

    MT 1804 – LINEAR ALGEBRA

 

 

 

Date : 28/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Answer ALL Questions.

  1. a) i) Let T be the linear operator on which is represented in the standard ordered

basis by the matrix .  Find a basis of, each vector of which is a

characteristic vector of T.

Or

  1. ii) Let T be a linear operator on a finite dimensional vector space V. Let

be the distinct characteristic values of T and let be the null space of

(T-I). If W=then prove that

dim W=.                                                 (5)

 

  1. b) i) State and prove Cayley-Hamilton theorem

Or

  1. ii) Let V be a finite dimensional vector space over F and T be a linear operator on V

then prove that T is triangulable if and only if the minimal polynomial for T is a

product of linear polynomials over F.                                                  (15)

 

  1. a) i) Let V be a finite dimensional vector space. Let be subspaces of V

let Then prove that the following are equivalent :

1)  are independent.

2) For each j,,= {0}

Or

iii)  Let T be a linear operator on a finite dimensional vector space V and let

are linear operators on V such that 1) each is a projection

2) 3) and let is the range of.  If each

is invariant under T then prove that T=T, i=1,2,..k.                      (5)

  1. b) i) Let T be a linear operator on a finite dimensional vector space V.  Suppose that

the minimal polynomial for T decomposes over F into a product of linear

polynomials.  Then prove that there is a diagonalizable operator D on V and

nilpotent operator N on V such that 1)  T= D+ N

2)  DN=ND

Or

  1. ii) Let T be a linear operator on a finite dimensional vector space V. Then

prove that T has a cyclic vector if only if the minimal and characteristic

polynomial for T are identical.                                                                     (15)

 

 

III. a) i)   Let T be a linear operator on which is represented in standard ordered basis

by the matrix . Prove that T has no cyclic vector.  What is the

T-cyclic subspace generated by the vector (1,-1,3)?

Or

  1. ii) If U is a linear operator on the finite dimensional vector space W and if U has a

cyclic vector then prove that there is an ordered basis for W in which U is

represented by the companion matrix of the minimal polynomial for U.     (5)

 

  1. b) i)  State and prove Cyclic Decomposition theorem.

Or

  1. ii) If T is a nilpotent operator on a vector space V of dimension n then prove that

characteristic polynomial for T is                                                          (15)

 

  1. a) i) Let V be a finite dimensional complex inner product space and f a form on V.

Then prove that there is an orthonormal basis for V in which the matrix of f is

upper-triangular.

Or

  1. ii) Let T be a linear operator on a complex finite dimensional inner product space
  2. Then prove that T is self-adjoint if and only if is real for every  in V.                                                                                                                                                                                                                                            (5)

 

  1. b) i) Let f be the form on defined by f=.Find the

matrix of f with respect to the basis {(1,-1),(1,1)}.

  1. ii) State and prove the spectral theorem. (6+9)

Or

iii) Let f be a form on a real or complex vector space V and a basis

for the finite dimensional subspace W of V.  Let M be the rxr matrix with

entries and W the set of all vectors in V such that

f ()=0 for all  W. Then prove that W is a subspace of V and

={0} if and only if M is invertible and when this is the case V=W+W.

(15)

  1. a) i) Let F be a field. Find all bilinear forms on the space .

Or

 

  1. ii) State and prove polarization identity for symmetric bilinear form f. (5)

 

  1. b) i)  Let V be a finite dimensional vector space over the field of complex numbers.

Let f be a symmetric bilinear form on V which has rank r. Then prove that there

is an ordered basis for V such that the matrix of f in the

ordered basis B is diagonal and f () =

Or

  1. ii) If f is a non-zero skew-symmetric bilinear form on a finite dimensional vector

space V then prove that there exist a finite sequence of pairs of

vectors,with the following properties:

1) f ()=1, j=1,2,,…,k.

2)  f ()=f ()=f ()=0,   ij.

3)  If is the two dimensional subspace spanned by  and , then

V=where is orthogonal to all and  and the restriction

of f to  is the zero form.                                                                                  (15)

 

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Loyola College M.Sc. Mathematics Nov 2008 Linear Algebra Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 26

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – November 2008

    MT 1804 – LINEAR ALGEBRA

 

 

 

Date : 04-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Answer ALL the questions.

 

  1. a) i) Prove that the similar matrices have the same characteristic polynomial.

OR

  1. ii) Let T be a linear operator on which is represented in the standard ordered basis by the matrix . Prove that T is not diagonalizable. (5)
  2. b) i) State and prove Cayley-Hamilton theorem.

OR

  1. ii) Let V be a finite dimensional vector space over F and T a linear operator on V.Then prove that T is diagonalizable if and only if the minimal polynomial for T has the form p=where are distinct elements of F. (15)

 

  1. a) i) Let V be a finite dimensional vector space. Let be subspaces such that with for. Then prove that are independent subspaces.

OR

  1. ii) Let W be an invariant subspace for T. Then prove that the minimal polynomial for divides the minimal polynomial for T.                   (5)
  2. b) i) State and prove Primary Decomposition theorem.

OR

  1. ii) If, then prove that there exist k linear operators on V such that
  2. Each is a projection.
  3. .

3.

  1. The range of is.

iii) Prove that if  are k linear operators which satisfy conditions 1, 2 and 3 of the above and if let be range of then.                                                                                                                       (8+7)

  • a) i) Let T be a linear operator on a vector space V and W a proper T-admissible subspace of V. Prove that W and Cyclic subspace Z(a;T) are independent.

OR

  1. ii) Let T be a linear operator on which is represented in the standard ordered basis by the matrix . Prove that T has no cyclic vector. What is the T-cyclic subspace generated by the vector (1,-1, 3)?                                           (5)
  2. b) i) Let a be any non-zero vector in V and let be the T-annihilator of . Prove the following statements:
  3. The degree of is equal to the dimension of the cyclic subspace      Z(a;T).
  4. If the degree of is k, then the vectors a, Ta, ,… form the   basis for Z(a;T).
  5. If U is the linear operator on Z(a;T) induced by T, then the minimal polynomial for U is .

OR

 

 

 

 

  1. ii) Let T be a linear operator on a finite dimensional vector space V and let

be a proper T-admissible subspace of V. Prove that there exist non-zero vectors in V with respective T-annihilators such that V=and divides, k=2, 3…r.                                    (15)

 

  1. a) i) Define a positive matrix. Verify that the matrix is positive.

OR

  1. ii) Let V be a complex vector space and f a form on V such that f () is real for every. Then prove that f is hermitian. (5)
  2. b) i) Let F be the field of real numbers or complex numbers. Let A be an nxn matrix over F. Then prove that the function g defined by is a positive form on the space if and only if there exists an invertible nxn matrix P with entries in F such that.
  3. ii) State and prove Principle Axis theorem. (6+9)

Or

iii) State and prove Spectral theorem and hence prove if, then for.                                                                             (15)

 

  1. a) i) Define a bilinear form on a vector space over a field. Let m and n be positive integers and F a field. Let V be the vector space of all mxn matrices over F and A be a fixed mxm matrix over F. If, prove that is a bilinear form.

Or

  1. ii) State and prove polarization identity for symmetric bilinear form f. (5)
  2. b) i) Let V be a finite dimensional vector space over the field of complex numbers. Let f be a symmetric bilinear form on V which has rank r. Then prove that there is an ordered basis for V such that the matrix of f in the ordered basis B is diagonal and f () =1, j=1,…,r. Furthermore prove that the number of basis vectors for which =1 is independent of the choice of basis.

Or

  1. ii) If f is a non-zero skew-symmetric bilinear form on a finite dimensional vector space V then prove that there exist a finite sequence of pairs of vectors,with the following properties:

1) f ()=1, j=1,2,,…,k.

2) f ()=f ()=f ()=0,ij.

3) If is the two dimensional subspace spanned by and, then V=where is orthogonal to all and  and the restriction of f to  is the zero form.                                                           (15)

 

 

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Loyola College M.Sc. Mathematics April 2009 Linear Algebra Question Paper PDF Download

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               LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 39

FIRST SEMESTER – April 2009

MT 1810 / 1804 – LINEAR ALGEBRA

 

 

 

Date & Time: 17/04/2009 / 1:00 – 4:00       Dept. No.                                                         Max. : 100 Marks

 

 

 

Answer ALL the questions.

 

  1. a) i) Let T be a linear operator on a finite dimensional space V and let c be a scalar. Prove that the following statements are equivalent.
  2. c is a characteristic value of T.
  3. The operator (T–cI) is singular.
  4. det (T–cI) =0.

OR

  1. ii) Let T be a linear operator on which is represented in the standard ordered basis by the matrix A=. Prove that T has no characteristic values in R. (5)
  2. b) i) Let T be a linear operator on a finite dimensional vector space V. Prove that the minimal polynomial for T divides the characteristic polynomial for T.

OR

  1. ii) Let V be a finite dimensional vector space over F and T be a linear operator on V then prove that T is triangulable if and only if the minimal polynomial for T is a product of linear polynomials over F. (15)

 

  1. a) i) Let V be a finite dimensional vector space. Let be independent subspaces such that , then prove that for.

OR

  1. ii) Let W be an invariant subspace for T. Then prove that the minimal polynomial for divides the minimal polynomial for T. (5)
  2. b) i) State and prove Primary Decomposition theorem.

OR

  1. ii) Let T be a linear operator on a finite dimensional space V. If T is diagonalizable and if are the distinct characteristic values of T, then prove that there exist linear operators on V such that

1..

  1. .
  2. .
  3. Each is a projection
  4. The range of is the characteristic space for T associated with.

iii) If there exist k distinct scalars and k non-zero linear operators which satisfy conditions 1,2 and 3, then prove that T is diagonalizable , are the distinct characteristic values of T and conditions 4 and 5 are satisfied also.                                                   (15)

 

  • a) i) Let T be a linear operator on a vector space V and W a proper T-admissible subspace of V. Prove that W and Cyclic subspace Z(a;T) are independent.

 

OR

  1. ii) If U is a linear operator on a finite dimensional space W, then prove that U has a cyclic vector if and only if there is some ordered basis for W in which U is represented by the companion matrix of the minimal polynomial for U. (5)

 

 

 

  1. b) i) ) Let a be any non-zero vector in V and let be the T-annihilator of . Prove the following statements:
  2. The degree of is equal to the dimension of the cyclic subspace      Z(a;T).
  3. If the degree of is k, then the vectorsa, Ta, ,… form the   basis for Z(a;T).
  4. If U is the linear operator on Z(a;T) induced by T, then the minimal polynomial for U is .

OR

  1. ii) Let T be a linear operator on a finite dimensional vector space V and let

be a proper T-admissible subspace of V. Then prove that there exist non-zero vectors in V with respective T-annihilators such that V=and divides, k=2, 3…r.                                                                                                                          (15)

 

  1. a) i) Define the matrix of a form on a real or complex vector space with respect to any ordered basis . Let f be the form ondefined by Find the matrix of f with respect to a basis {(1,-1), (1, 1)}.

OR

  1. ii) Let T be a linear operator on a complex finite dimensional inner product space V. Then prove that T is self-adjoint if and only if is real for every in V.                                                                             (5)
  2. b) i) Let f be the form on a finite-dimensional complex inner product space V. Then prove that there is an orthonormal basis for V in which the matrix of f is upper-triangular.
  3. ii) Prove that for every Hermitian form f on a finite-dimensional inner product space V, there is an orthonormal basis of V in which f is represented by a diagonal matrix with real entries.        (6+9)

OR

iii) Let f be a form on a real or complex vector space V and a basis for the finite dimensional subspace W of V. Let M be the r x r matrix with entries and Wthe set of all vectors  in V such that

f ()=0 for all in W. Then prove that Wis a subspace of V,={0} if and only if M is invertible and when this is the case V=W+W.                                                                                          (15)

  1. a) i) Let V be a vector space over the field F. Define a bilinear form f on V and

prove that the function defined by f () =LLis bilinear.

OR

  1. ii) Define the quadratic form q associated with a symmetric bilinear form f and prove that . (5)
  2. b) i) Let V be a finite dimensional vector space over the field of complex numbers.Let f be a symmetric bilinear form on V which has rank r. Then prove that there is an ordered basis for V such that the matrix of f in the ordered basis B is diagonal and f () =

OR

  1. ii) If f is a non-zero skew-symmetric bilinear form on a finite dimensional vector space V then prove that there exist a finite sequence of pairs of vectors,with the following properties:

1) f ()=1, j=1,2,,…,k.

2) f ()=f ()=f ()=0,ij.

3) If is the two dimensional subspace spanned by and, then V=where is orthogonal to all and  and the restriction of f to  is the zero form.                                              (15)

 

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Loyola College M.Sc. Mathematics Nov 2010 Linear Algebra Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2010

MT 1810/ 1804 – LINEAR ALGEBRA

 

 

 

Date : 30-10-2011             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

                                  

  1. a) (i) Prove that similar matrices have the same characteristic polynomial.

 

(OR)                                                                                        (5)

 

(ii) Let T be the linear operator on Â3 which is represented in the standard ordered basis by

the matrix .  Find the characteristic polynomial of A.

  1. b) (i) State and prove Cayley-Hamilton theorem.

 

(OR)                                                                                           (15)

(ii) Let V be a finite dimensional vector space over F and T a linear operator on V.  Then

prove that T is diagonalizable if and only if the minimal polynomial for T has the

form are distinct elements of F.

  1. a) (i) Let T be a linear operator on a finite dimensional space V and let c be a scalar. Prove

that the following statements are equivalent.

  1. c is a characteristic value of T.
  2. The operator (T – cI) is singular.
  3. det (T – cI) = 0.

(OR)                                                                                        (5)

 

(ii) Let W be an invariant subspace for T.  Then prove that the minimal polynomial for Tw

divides the minimal polynomial for T.

 

  1. b) (i) State and prove Primary Decomposition theorem.

(OR)                                                                                           (15)

 

(ii) Let T be a linear operator on a finite dimensional space V.  If T is diagonalizable and if

c1,…,ck  are the distinct characteristic values of T, then prove that there exist linear

operators E1,…,Ek on V such that

  1. T = c1E1 +…+ ckEk.
  2. I = Ej +…+ Ek.
  3. EiEj = 0,i≠j.
  4. Each Ei is a projection

III. a)  (i) Let W be a proper T-admissible subspace of V. Prove that there exists a nonzero a in

V such that W Ç Z (a ; T) = {0}.

(OR)                                                                                             (5)

 

(ii) Define T– annihilator, T–-admissible, Projection of vector space V and Companion

matrix.

 

  1. b) (i) State and prove Cyclic Decomposition theorem.

 

(OR)                                                                                           (15)

 

(P.T.O.)

  1. ii) Let P be an m x m matrix with entries in the polynomial algebra F [x]. The following are

equivalent.

 

  • P is invertible
  • The determinant of P is a non-zero scalar polynomial.
  • P is row-equivalent to the m x m identity matrix.
  • P is a product of elementary matrices.

 

  1. a) (i) Let V be a complex vector space and f be a form on V such that f (a,a) is real for

every a.  Then prove that f is Hermitian.                                                                                          (5)

(OR)

  1. ii) Let f ­ be the form on a finite-dimensional complex inner product space V. Then prove

that there is an orthonormal basis for V in which the matrix of f is upper-triangular.

 

  1. b) (i) Let f be a form on a finite dimensional vector space V and let A be the matrix of f in an

ordered basis B. Then f is a positive form iff A = A* and the principal minors of A are all

positive.

(OR)                                                                         (15)

(ii) Let V be a finite-dimensional inner product space and f a form on V.  Then show that there is a

unique linear operator T on V such that f(a,b) = (Ta½b) for all a, b in V, and the map f ®T is an

isomorphism of the space of forms onto L(V,V).

 

 

 

  1. a) (i) Let V be a vector space over the field F. Define a bilinear form f on V and prove that the

function defined by f (a;b) = L1 (a) L2 (b) is bilinear.

 

(OR)                                                                            (5)

 

  1. ii) Define the quadratic form q associated with a symmetric bilinear form f and prove that

                                                                                                            

 

  1. b) i) Let V be a finite dimensional vector space over the field of complex numbers. Let f be a symmetric bilinear form on V which has rank r.  Then prove that there is an ordered basis

B ={b1, b2, … bn} for V such that the matrix of f in the ordered basis B is diagonal and  .

(OR)                                                                                (15)

  1. ii) If f is a non-zero skew-symmetric bilinear form on a finite dimensional vector space V then

prove that there exist a finite sequence of pairs of vectors, (a1, β1), (a2, β2),… (ak, βk) with the

following properties.

  1. f (aj, βj) = 1 , j=1,2,,…,k.
  2. f (ai, aj)=f(βi, βj­)=f(ai,βi)=0,i≠j.
  3. c) If Wj is the two dimensional subspace spanned by aj and βj, then V=W1 Å W2Å …Wk Å W0

where W0 is orthogonal to all aj and βj and the restriction of f  to W0 is the zero form.

 

 

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Loyola College B.Sc. Mathematics April 2008 Linear Algebra Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 20

 

FIFTH SEMESTER – APRIL 2008

MT 5502 – LINEAR ALGEBRA

 

 

 

Date : 03/05/2008                Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

SECTION – A                                  

Answer ALL the questions.                                                             (10 x 2 = 20 marks)

 

  1. Illustrate by an example that union of two subspaces need not a subspace of a vector space.
  2. Give an example of a linearly dependent set of vectors in R2 over R.
  3. “Every linearly independent set of vectors is a basis.” True or False. Justify.
  4. Show that kernel of a homomorphism in a vector space is a subspace.
  5. If V is an inner product space show that where a and b are scalars.
  6. For T є A(v)define eigen value and eigen vector of T.
  7. Find the trace of
  8. Is the following matrix skew symmetric?
  9. If T є A(v)is a Hermitian show that all its eigen values are real.
  10. When do you say that a linear transformation T on V is unitary?

 

SECTION – B                                  

Answer any FIVE questions.                                               (5 x 8 = 40 marks)

 

  1. Let V be a vector space of dimension n and  be libearly independent vectors in V. Show that there exists n-r new vectors in V such that is a basis of V.
  2. If V is a vector space of finite dimension that is the direct sum of its subspace U and W show that
  3. Verify that defined by T(a,b)=(a-b, b-a, -a) for a, b є R is a vector space homomorphism. Find the rank and nullity of T.
  4. State and prove triangular inequality.
  5. Show the T є A(V)is invertible if and only if the constant term of the minimal polynomial for T is not zero.
  6. If V has dimension n andT є A(V), show that the rank of T is equal to the rank of the corresponding matrix m(T) in Fn.

 

 

  1. Check the consistency of the following set of equations.

x1+2x2+x3+11

4x1+6x2+5x3=8

2x1+2x2+3x3=19

  1. If for all v є V, show that T is unitary.

SECTION – C

Answer any TWO questions.                                               (2 x 20 = 40 marks)

 

  1. (a) Find whether the vectors (2, 1, 1, 1), (1, 3, 1, -2) and (1, 2, -1, 3) in R4 are linearly dependent or independent.

(b) If U and V are vector spaces of dimension m and n respectively over F, show that the vector space Hom(U,V) is of dimension mn.

  1. a) Apply Gram Schmidt orthonormalization process to obtain an orthonormal basis for the subspace of R4 generated by the vectors. (1, 1, 0, 1), (1, -2, 0, 0),

(1, 0, -1, -2).

  1. b) If are distinct eigen values of T є A(V) and if v1, v2, …, vn are eigen vectors of T belonging respectively, show that v1,v2,…,vn are linearly independent over F.
  1. a) Let V be a vector space of dimension n over F and let T є A(V). If m1(T) and m2(T) are the matrices of T relative to two bases {v1, v2, …, vn} and {w1,w2,…,wn} of V respectively, show that there is an invertible matrix C in Fn such that m2(T)=Cm1(T)C-1.
  1. b) Show that any square matrix A can be expressed uniquely as the sum of a symmetric matrix and a skew-symmetric matrix.
  2. c) If A and B are Hermitian show that AB+BA is Hermitian and AB-BA is skew Hermitian.
  1. a) Find the rank of
  1. b) If T є A(V)show that T* є A(V)and show that

(i)

(ii)  

(iii) (T)* = T*

(iv) (T*)* = T

 

 

 

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Loyola College B.Sc. Mathematics Nov 2008 Linear Algebra Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

QB 02

 

FIFTH SEMESTER – November 2008

MT 5502 – LINEAR ALGEBRA

 

 

 

Date : 10-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

Answer all questions:                                                                        (10 x 2 = 20 marks)

 

  1. Show that union of two subspaces of V need not be a subspace of V.
  2. If the set {v1, v2, ….vm} is a linearly independent set of vectors of a vector space V then prove that any non-empty set of this set is linearly independent.
  3. Define basis of a vector space and give an example.
  4. Show that  defined by  is a vector space homomorphism.
  5. Define an inner product space and give an example.
  6. Define an Eigen value and an Eigen vector.
  7. If {vi} is an orthonormal set, then prove that the vectors in {vi} are linearly independent.
  8. If A and B are Hermitian, Show that AB_BA is skew-Hermitian.
  9. Let R3 be the inner product space over R under the standard inner product. Normalize .
  10. Prove that the product of two invertible linear transforms on V is itself an invertible linear transformation on V.

SECTION – B

 

Answer any FIVE questions:                                                           (5 x 8 = 40 marks)

 

  1. Show that } is a basis of the Vector space F[x] of all polynomials of degree at most n.
  2. If A and B are subspaces of a vector space V over F, prove that (A+B) / B A / AB.
  3. State and prove Schwarz’s inequality.
  4. Prove that  is invertible if and only if the constant term of the minimal polynomial for T is not zero.
  5. If dim V=n and , then prove that T can have atmost n distinct eigen values.
  6. Let V=R3 and suppose that  is the matrix of  relative to the standard basis (1,0,0), (0,1,0), (0,0,1). Find the matrix relative to ,  & .
  7. Let A be an mxn matrix over a field F, and let r be its rank. Then prove that is equal to the size of the largest non-singular square submatrix of A.
  8. If  in V, then prove that T is unitary.

 

SECTION – C

Answer any TWO questions:                                                           (2 x 20 = 40 marks)

 

  1. a)   If V is a vector space of finite dimension that is the direct sum of its subspaces U and

W,  then prove that .

  1. Find the Co-ordinate vector of (2, -1, 6) of R3 relative to the basis .                                                                   (15+5)
  2. If then prove that is of dimension m2.
  3. a) State and prove Gram-Schmidt ortho normalisation theorem.
  4. Normalize in C3, relative to the standard inner product. (15+5)
  5. a) Prove that is invertible if and only if the constant term in the minimal polynomial for T is not zero.
  6. b) The linear transformation T on V is unitary if and only if it takes an orthonormal basis of V onto an orthonormal basis of V. (14+6)

 

 

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Loyola College B.Sc. Mathematics April 2009 Linear Algebra Question Paper PDF Download

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          LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 32

FIFTH SEMESTER – April 2009

MT 5508 / 5502 – LINEAR ALGEBRA

 

 

 

Date & Time: 06/05/2009 / 1:00 – 4:00            Dept. No.                                               Max. : 100 Marks

 

 

SECTION – A

      Answer ALL the questions.                                                                          (10 X 2 = 20)

  1. Let V be a vector space.  Prove that (– a)v = a(– v) = – (av) for any a Î F and vÎ V.
  2. Prove that any non-empty subset of linearly independent vectors is linearly independent.
  3. Show that the vectors (1, 0, -1), (2, 1, 3), (–1, 0, 0) and (1, 0, 2) are linearly dependent in R3.
  4. Verify that the map T : R2 ® R2 defined by T(x, y) = (2x + y, y) is a homomorphism.
  5. If V is an inner product space, then prove that
  6. Define eigenvalue and eigenvector of a linear transformation.
  7. If l Î F is an eigenvalue of T Î A(V), prove that l is a root of the minimal polynomial of T over F.
  8. For A, B Î Fn, prove that tr (A + B) = tr (A) + tr (B).
  9. If A and B are Hermitian, show that AB + BA is Hermitian.
  10. Prove that the eigenvalues of a unitary transformation of T are all of absolute value 1.

 

SECTION – B

Answer any FIVE questions only.                                                         (5 X 8 = 40)

  1. Prove that the union of two subspaces of a vector space V over F is a subspace of V if and only if one is contained in the other.
  2. If V is a vector space of finite dimension and is the direct sum of its subspaces U and W, then prove that dim V = dim U + dim W.
  3. Let U and V be vector spaces over a field F, and suppose that U has finite dimension n. Let {u1, u2, . . . un} be a basis of U and let v1, v2, . . . vn be arbitrary vectors in V. Prove that there exists a unique homomorphism T: U ® V such that T(u1) = v1, T(u2) = v2, . . . , T(un) = vn.
  4. Define an orthonormal set.  If {w1, w2, . . . wn} is an orthonormal set in an inner product space V, prove that .
  5. Prove that T Î A(V) is invertible if and only if the constant term of the minimal polynomial for T is not zero.

 

  1. Let V = R3 and suppose that  is the matrix of T Î A(V) relative to the standard basis v1 = (1, 0, 0), v2 = (0, 1, 0), v3 = (0, 0, 1).  Find the matrix of T relative to the basis w1 = (1, 1, 0), w2 = (1, 2, 0) and w3 = (1, 2, 1).
  2. Show that any square matrix can be expressed uniquely as the sum of a symmetric matrix and a skew-symmetric matrix.
  3. Investigate for what values of l, m, the system of equations  x1 + x2 + x3 = 6,
     x1 + 2x2 + 3x3 = 10,   x1 + 2x2 + l x3 = m  over a rational field has (a) no solution  (b) a unique solution  (c) an infinite number of solutions.

 

SECTION – C

Answer any Two questions.                                                                   (2 X 20 = 40)

  1. (a) Let T : U ® V be a homomorphism of two vector spaces over F and suppose that U has finite dimension. Then prove that

                              Dim U = nullity T + rank of T

(b)  If W1 and W2 are subspaces of a finite dimensional vector space V, prove that dim (W1 + W2) = dim W1 + dim W2 – dim (W1 Ç W2)

  1. (a) Prove that T Î A(V) is singular if and only if there exists an element v ¹ 0 in V such that T(v) = 0.

(b)  Prove that every finite-dimensional inner product space V has an orthonormal set as a basis.    (8+12)

  1. (a) If T be the linear transformation on R3 defined by
    T(a1, a2, a3) = ( a1 + a2 + a3, – a1 – a2 – 4a3, 2a1 – a3), find the matrix of T relative to the basis v1 = (1, 1, 1), v2 = (0, 1, 1), v3 = (1, 0, 1).

(b)  Prove that the vector space V over F is a direct sum of two of its subspaces W1 and W2 if and only if V = W1 + W2 and W1 Ç W2 = (0).                                                                    (8+12)

  1. (a) If T Î A(V), then prove that T* Î A(V).  Also prove that

(i) (S+T)* = S* + T*     (ii)  (ST)* = T*S*     (iii)       (iv)  (T*)* = T

(b)  Prove that the linear transformation T on V is unitary if and only if it takes an orthonormal basis of V onto an orthonormal basis of V.

 

 

 

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Loyola College B.Sc. Mathematics April 2011 Linear Algebra Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2011

MT 5508/MT 5502 – LINEAR ALGEBRA

 

 

 

Date : 20-04-2011              Dept. No.                                                  Max. : 100 Marks

Time : 1:00 – 4:00

 

SECTION – A

Answer ALL questions                                                                                         (10 X 2 = 20 Marks)

 

  1. Define a vector space over a field F.
  2. Prove that R is not a vector space over C.
  3. Define the kernel of a linear transformation.
  4. Prove that in V3(R), the vectors (1, 2, 1), (2,1,0) and (1, -1, 2) are linearly independent.
  5. Define an inner product space.
  6. State the triangle inequality for inner product space.
  7. Define an orthonormal set in an inner product space.
  8. Prove that (A+B)T = AT + BT where A and B are two m X n matrices.
  9. Define an invertible matrix.
  10. Define Hermitian and unitary linear transformations.

SECTION – B

 

Answer any FIVE questions                                                                                (5 X 8 = 40 Marks)

 

  1. Prove that any set containing a linearly dependent set is also linearly dependent.
  2. Let V be a vector space over a field F. Then prove that S = {v1, v2, . . ., vn} is a basis for V if and only of every element of V can be expressed as a linear combination of elements of S.
  3. Prove that T : R2→R2 defined by T(a, b) = (a+b, a) is a vector space homomrphism.
  4. Prove that T Є A(V) is invertible if and only of T maps V onto V.
  5. Let T Є A(V) and λ Є F. Then prove that λ is an eigenvalue of T if and only if λI-T is singular.
  6. Show that any square matrix can be expressed uniquely as the sum of a symmetric and a skew – symmetric matrix.
  7. Show that the system of equations

X+2y+z=11

4x+6y+5z=8

2x+2y+3z=19 is inconsistent.

  1. If TЄA(V) is Hermitian, then prove that all its eigen values are real.

SECTION – C

 

Answer any TWO questions                                                                         (2 X 20 = 40 marks)

 

  1. a) If V is a vector space of finite dimension and is the direct sum of its subspaces U and

W, then prove that dim V = dim U + dim W.

 

  1. b) If A and B are subspaces of a vector space V over F, prove that (A+B)/BA/A

 

(10 x 10)

  1.    If U and V are vector spaces of dimensions m and n respectively over F, prove that

Hom (U,V) is of  dimension mn.

 

  1. a) Apply the Gram – Schmidt orthonormalization process to the vectors (1,0,1), (1,3,1)

and (3,2,1) to obtain an orthonormal basis for R3.

 

  1. b) State and prove Bessel’s inequality.                                             (10 + 10)

 

  1. a) Let V=R3 and suppose that is the matrix of T Є A(V) relative to the

standard basis V1 = (1,0,0), V2 = (0, 1, 0), V3 = (0,0,1). Find the matrix of T relative to

the basis W1 = (1,1,0),  W2 =  (1,2,0), W3 = (1,2,1).

 

  1. b) Show that the linear transformation T on V is unitary if and only if it takes an

orthonormal basis of V onto an orthonormal basis of V.                                     (10 + 10)

 

 

 

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Loyola College B.Sc. Mathematics April 2012 Linear Algebra Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2012

MT 5508/MT 5502 – LINEAR ALGEBRA

 

 

 

Date : 03-05-2012              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

Answer ALL questions:                                                                                          (10 x2 =20 marks)

  1. If V  is a vector over a field F, Show that (-a)v = a(-v)= -(av), for a  F, v
  2. Is the Union of two subspaces is a subspace?
  3. Show that the vectors (1,0,-1), (2,1,3),(-1,0,0) and (1,0,1) are linearly dependent in .
  4. Determine the following mapping is a vector space homomorphism: T :   by T(a,b)=ab.
  5. Define inner product space.
  6. Define orthonormal set in an inner product space.
  7. Prove that is orthogonal.
  8. For A,B Fn and then prove that tr (A+B) = tr A + tr B.
  9. Define Hermitian and skew-Hermitian.
  10. Find the rank of the matrix over field of rational numbers.

PART – B

 

Answer any FIVE questions:                                                                                   (5×8=40 marks)

 

  1. Prove that a non empty subset W of a vector space V over F is a subspace of V if and only if aw1+bw2 W  , for all a,b F , w1,w2
  2. If v1,v2,. . .,vn V are linearly independent , and if v V is not in their linear span, Prove that  v1,v2,. . .,vn  are linearly independent.
  3. Find the coordinate vector of (2,1,-6) of R3 relative to the basis {(1,1,2),(3,-1,0),(2,0,-1)}.
  4. Prove that T :   defined by T(a,b) = (a-b, b-a,-a) for all a,b is a vector space homomorphism.
  5. State and Prove Schwarz inequality.
  6. If is an eigen value of T  A(v), then for any polynomial f(x)  F[x] , f() is an eigen value of f(T).
  7. If A and B are Hermitian , Show that AB + BA is Hermitian and AB-BA is skew-Hermitian.
  8. Show that the system of equations x1+2x2+x3 = 11, 4x1+6x2+5x3 = 8,  2x1+2x2+3x3 = 19 is inconsistent.

PART – C

Answer any TWO questions:                                                                                         (2 x20 =40 marks)

 

  1. (a) Let V be a vector space of finite dimension and let  W1 and W2 be subspaces of V such that

 

V = W1+W2 and dim V = dim W1+dim W2. Then  prove that V = W1W2.                                         (10 + 10)

 

(b) If A and B are subspaces of a vector space V over F , Prove that

 

  1. (a) If V is a finite dimensional inner product space and w is a sub space of V, prove that .

 

(b) Show that is invertible if and only if the constant term of the minimal polynomial

for T is  not zero.                                                                                                              (10 + 10)

 

21.(a) If are distinct eigen values of   and if v1,v2,. . .,vn are eigen vectors of

T belonging to   respectively, then v1,v2,. . .,vn are linearly independent over F.

 

(b) If A,B Fn, where F is the complex field, then

(i) ,    (ii)  , ,    (iii) ,    (iv) .

(10 + 10)

 

 

  1. (a) The linear transformation T on V is unitary if and only if it takes an orthonormal basis of V onto

an orthonormal basis of  V.

 

(b) (i) If  is skew-Hermitian , Prove that all of its eigen  values are pure imaginaries.

(ii) Prove that the eigen values of a unitary transformation are all of absolute value one.

 

(10 + 10)

 

 

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Loyola College B.Sc. Mathematics Nov 2012 Linear Algebra Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – NOVEMBER 2012

MT 5508/MT 5502 – LINEAR ALGEBRA

 

 

Date : 08/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

 

Answer ALL questions:                                                                                           (10 x 2 = 20 marks)

 

  1. Define a vector space over a field F.
  2. Show that the vectors (1,1) and (-3, 2) in R2 are linearly independent over R, the field of real numbers.
  3. Define homomorphism of a vector space into itself.
  4. Define rank and nullity of a vector space homomorphism T: u®
  5. Define an orthonormal set.
  6. Normalise in R3 relative to the standard inner product.
  7. Define a skew symmetric matrix and give an example.
  8. Show that is orthogonal.
  9. Show that is unitary.
  10. Define unitary linear transformation.

 

PART – B

 

Answer any FIVE questions:                                                                             (5 x 8 =40 marks)

 

  1. Prove that the intersection of two subspaces of a vector space v is a subspace of V.
  2. If S and T are subsets of a vector space V over F, then prove that
  3. S T implies that L(S) ≤ L(T)
  4. L(L(S)) = L(S)
  5. L(S U T) = L(S) + L(T).
  6. Determine whether the vectors (1,3,2), (1, -7, -8) and (2, 1, -1) in R3 are linearly dependent on independent over R.
  7. If V is a vector space of finite dimension and W is a subspace of V, then prove that

dim V/W = dim V – dim W.

  1. For any two vectors u, v in V, Prove that .
  2. If and l Î F, then prove that l is an eigen value of T it and only if [l I – T] is singular.
  3. Show that any square matrix can be expressed as the sum of a symmetric matrix and a skew symmetric matrix.
  4. For what values of T, the system of equations over the rational field is consistent?

PART – C

 

Answer any TWO questions:                                                                    (2 x 20 = 40 marks)

 

  1. a) Prove that the vector space V over F is a direct sum of two of its subspaces W1 and W2

if and only if V = W1 + W2 and W1  W2 = {0}.

 

  1. b) If V is a vector space of finite dimension and is the direct sum of its subspaces U and

W, than prove that dim V = dim U + dim W.                                                                   (10 + 10)

 

  1. If U and V are vector spaces of dimension m and n respectively over F, then prove that the

vector space Hom (U, V) is of dimension mn.

 

  1. Apply the Gram – Schmidt orthonormalization process to obtain an orthonormal basis for

the subspace of R4 generated by the vectors (1, 1, 0, 1) , (1, -2, 0, 0) and (1, 0, -1, 2).

 

  1. a) Prove that TÎA(V) is singular if and only it there exists an element v ≠ 0 in V such that

T(v) = 0.

  1. b) Prove that the linear transformation T on V is unitary of and only if it takes an

orthonormal basis of V onto an orthonormal basis of V.                                                       (10 +10)

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