LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS
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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.
- 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.
- c is a characteristic value of T.
- The operator (T–cI) is singular.
- det (T–cI) =0.
OR
- 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)
- 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
- 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)
- a) i) Let V be a finite dimensional vector space. Let be independent subspaces such that , then prove that for.
OR
- ii) Let W be an invariant subspace for T. Then prove that the minimal polynomial for divides the minimal polynomial for T. (5)
- b) i) State and prove Primary Decomposition theorem.
OR
- 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..
- .
- .
- Each is a projection
- 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
- 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)
- b) i) ) Let a be any non-zero vector in V and let be the T-annihilator of . Prove the following statements:
- The degree of is equal to the dimension of the cyclic subspace Z(a;T).
- If the degree of is k, then the vectorsa, Ta, ,… form the basis for Z(a;T).
- If U is the linear operator on Z(a;T) induced by T, then the minimal polynomial for U is .
OR
- 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)
- 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
- 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)
- 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.
- 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)
- 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
- ii) Define the quadratic form q associated with a symmetric bilinear form f and prove that . (5)
- 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
- 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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