BSc Mathematics SEM V 2018 19 2018-19 Maths Linear Algebra Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate marks for respective subquestions

1. Q1 (a) Answer any ONE
   • i. If : > is an isometry such that T(0) = 0 then, show that T — an orthogonal linear transformation 8
   • ii. state and prove the Cayley Hamilton theorem. 8
   • (b) Answer any TWO
   • i. Let V be a finite dimensional real vector space and W be of V. Show that dimV/W 6
   • ii. Let A = 0 A linear transformation T R? is defined by = AX(X being a column vector in R*). Find kerT,a basis of kerT and Also find ImT
   • iii. Show that similar matrices have same characteristic polynomial
   • iv. If is a nilpotent real matrix then prove that = 0 for all positive 6
2. Q2 (a) Answer any ONE
   • i. Show that a real matrx with n eigen values is similar to an upper triangular matrix of order n with the eigen values on diagonal 8
   • ii. Show that minimal polynomial of a real matrix divides every polynomial which annihilates A. Further prove that a is a root of the minimal polynomial of matrix A if and only if a is a characteristic root 8
   • (b) Answer any TWO
   • i. Show that eigen vectors Corresponding to distinct eigen values respectively of a square matrix A are linearly in 6
   • ii. If A,B are n x n real matrices and A is non-singular then show that AB and BA have same eigen values
   • iii. Let be a real matrix and 1, -1, 3 be its eigen values. Which of A, A?—A, A?+3A, matrices are nonsingular? Justify your answer
   • iv. Let be a real matrix. if A has n distinct characteristic roots, then prove that the characteristic polynomial of A = the minimal polynomial 6
3. Q3 (a) Answer any ONE
   • i. Show that an n x n matrix A is diagonalizable if and only if A has n eigen values where algebraic multiplicity of each eigen value coincides with its geometric multiplicity
   • ii. Show that a quadratic form Q is positive definite if and only if all eigen values of associated symmetric matrix are positive 8
   • (b) Answer any TWO
   • i. Show that a real matrix with distinct eigen values is diagonaliz
   • ii. Let A symmetric matrix. Show that (AX,Y) forall X,Y R”. Hence or otherwise, prove that eigen vectors correspond ing to distinct eigen values of a real symmetric matrix are mutually
   • iii. If A is an x n diagonalizable matrix with eigen values 1 and —1, show
   • iv. Show that a non-zeron xn matrix of rank 1 is diagonalisable with eigen 4, Answer any THREE
   • (a) Find the orthogonal transformations in which represent reflection with 5
   • (b) Show that a : > defined by = + BY — 2, — is an isometries. Express it as a composite of an orthogonal transformation and a translation 5
   • (c) Let V be a vector space of finite dimension and V V bea linear transformation. Show that eigen space corresponding to any eigen value of T is invariant under
   • (d) Let be a matrix with all the entries as 1. Find eigen values and the corresponding eigen spaces of A (ec) Prove that if every non-zero vector of is an eigen vector of then
   • (f) By applying rotation of coordinate axes reduce the conic = standard form. Hence identify it 5

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