BSc Mathematics SEM V 2017 18 2017-18 Mathematics 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. Let V be a finite dimensional inner product space. If f — V is a function such that (p) f(0) = 0 (q) = X,Y then show that f is an orthogonal linear transformation 8
   • ii. State and prove the Cayley-Hamilton theorem. 8
   • (b) Answer any TWO
   • i. Let W be a subspace of a finite dimension real vector space V. Show that 6
   • ii. Show that a 2 x 2 orthogonal matrix with determinant = 1 is a matrix of 6
   • iii. Find an orthogonal transformation in R? which represents reflection with respect to the plane x —y+ 0 6
   • iv. If T : R? —> R? is a linear transformation such that < u,v > => < T(u),T(v) > R? then show that T where a R and — is an orthogonal transformation 6
2. Q2 (a) Answer any ONE
   • i. Define the minimal polynomial of a square matrix A. Show that - 8
   • (p) the minimal polynomial of a real matrix A divides every polynomial
   • (q) A is a root of the minimal polynomial of A if and only if \ is a charac teristic root of A
   • ii. Let A be an n x n matrix having n eigenvalues then prove that A is similar to an upper triangular matrix with the n eigen values on the diagonal of the 8
   • (b) Answer any TWO
   • i. Let be a real matrix. Show that A is an eigenvalue of A if and only if — A) is singular. Hence or otherwise show that 0 is not an eigen value of an injective linear transformation : > R” 6
   • ii. Let A and B be n x n real matrices. Prove that characteristic polynomial of AB = characteristic polynomial of BA ili. Find the characteristic polynomial and the minimal polynomial of | 2 1 0 | . (6) 6
   • iv. Prove that if every non-zero vector of R” is an eigenvector of A, then A is a 6
   • Q. P. Code: 19566
3. Q3 (a) Answer any ONE
   • i. Show that an nxn matrix A is diagonalizable if and only of dimensions of eigen spaces of A is n 8
   • ii. Show that characteristic roots of a real symmetric matrix are real. 8
   • (b) Answer any TWO
   • i. Let A be any n x n diagonalizable matrix. Show that 6
   • (p) For any positive integer k, A* is also diagonalizable
   • (q) f(A) is diagonalizable where f(t) is any polynomial over R
   • ii. Show that every quadratic form over R can be reduced to standard form Ay? by an orthogonal change of variables X = PY,X = y = and n orthogonal matrix Let A = Find a non-singular matrix P such that is a (6) diagonal matrix and hence find 6
   • iv. Find the rotation of coordinate axes which reduces the conic x? + + to standard form. Give its equation in the standard form in the rotated system and identify the conic 6
4. Q4 Answer any THREE
   • (a) Let be an upper triangular matrix whose diagonal entries are 1, 2 , 3. If =aA?+bA-+cl, then find a,b, c 5
   • (b) If a : > R* given by a(z,y) = (ax + by +f) for real numbers f isan isometry, then prove that a?+c? = 1, b?+d? = 1 and ab+cd = 0 5
   • (c) Find the eigenvalues and the bases of the corresponding eigen spaces for a 3 x 3 matrix A having all its entries equal to 5
   • (d) Let V be a finite dimensional inner product space over R and V V be a linear transformation. Define an invariant subspace under 7’. Prove that for 5
   • (i) Ey ={X is a subspace of V
   • (ii) is an invariant subspace under Let A and B be positive definite matrices. Show that A+ B is also positive (5) definite. Is the converse true? Justify your answer
   • (f) Find the rank and signature of 5

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