BSc Mathematics SEM V ATKT May 2018-19 ATKT 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 Fill in the blank by choosing the correct option
   • i. Consider W = 22 +2y+2= 2
   • (a) (b) (c) (d) None of these
   • (a) an orthogonal matrix of reflection
   • (b) an orthogonal matrix of rotation
   • (c) not an orthogonal matrix
   • (d) None of these
   • iii. If = t? is characteristic polynomial of A and po(t)= t? + bjt + bo is characteristic polynomial of A? then b)
   • (a) (b) None of these
   • iv. 0 is an eigen value of a linear transformation
   • (a) invertible (b) not invertible
   • (c) None of these
   • v. Let A = be a 10 x 10 matrix with aij = Then the set of eigen values of A is ———
   • (c) (d) None of these
   • vi. Let R? — be the orthogonal transformation of reflection in a straight line passing through origin then 7’ has ——— eigen
   • (a) only (b) only (c) (d) None of these
   • vii. The minimal polynomial of for any a 4 0 is 2
   • (a) (b) (c) None of these
   • viii. Let A and B be 3 x 3 non-diagonal matrices over R such that
   • (c) B are diagonalisable. (d) None of these
   • (a) diagonalisable but not orthogonally diagonalisable
   • (b) orthogonally diagonalisable
   • (c) not diagonalisable
   • (d) None of these
   • x. The rank and signature of the quadratic from = 2
   • (c) None of these
2. Q2 (a) Answer any ONE
   • i. Let V be an n dimensional inner product space and W be a subspace of V of dimension n 1. be a unit vector orthogonal to W. Show that V V defined by T(x) = x is an orthogonal linear transformation such that il. State and prove the Cayley Hamilton theorem. (8)
   • (b) Answer any TWO
   • i. Let V be a finite dimensional inner product space and f : V > V be an isometry, then show that there exists unique V and an unique orthogonal linear transformation V > V such that f = where L,,: V — V is a translation map defined as L,,(X) = X + Xo 6
   • ii. Let V be a finite dimensional inner product spaace and V V be alinear transformation. Prove that T is orthogonal if and only if = VX V
   • iii. If matrix has the characteristic polynomial x? + 2x — 1, then find the value of det A) 6
   • iv. If w is a unit column vector in and A= Then prove that A is an orthogonal matrix 6
3. Q3 (a) Answer any ONE
   • i. If is an eigen value of a real n x n matrix A, then 8
   • (p) A is an eigen value of A’
   • (q) is an eigen value of A* for k N. Hence f(A) is an eigen value of f(A), for a polynomial over R
   • (r) If A is invertible, then \~! is an eigen value of
   • ii. Define the minimal polynomial of a square matrix A. Prove that similar matrices have same minimal polynomials. Is the 8
   • (b) Answer any TWO
   • i. Let Anxn be a real matrix. Show that eigen vectors corre- sponding to distinct eigen values, , Ax, of A are lin 6
   • ii. Define invariant subspace. Let V be a finite dimension vector space and V V be a linear transformation. Show that space of is invariant under 7’
   • iii. Find the characteristic polynomial and the minimal polyno- 6
   • iv. Let A be a 13 x 13 real matrix of rank 1 and P(t) be the characteristic polynomial of A then prove that P(t) = —
4. Q4 (a) Answer any ONE
   • i. Show that real symmetric matrix of order n is orthogonally
   • ii. Define algebraic and geometric multiplicities of an eigen of a square matrix. Show that the geometric multiplicity of an eigen value does not exceeds its algebraic multiplicity 8
   • (b) Answer any TWO
   • i. Show that areal n x n is diagonalisable if and only if basis of IR” consisting of eigen vectors of A
   • ii. Let A be an n x n real symmetric matrix. Then show that > 0 for all non-zero X R” if and only if each eigenvalue of A is positive
   • iv. Let A3,3 real matrix having 1, -1 3 as eigen values. Determine which of the matrices in S are non-singular where = {A? + A, A? — A, A? +3A, A? — 3A}. Justify your answer 6
5. Q5 Answer any FOUR
   • (a) Prove or disprove: If the characteristic polynomial of a matrix Anxn iS same as minimal polynomial then A has distinct eigen 5
   • (b) Find an orthogonal transformation in R* which represents reflec tion with respect to the plane x 0
   • (c) Find the eigenvalues and the bases of the corresponding eigen
   • (d) Let A2 be the distinct eigenvalues of A with as corre sponding eigen vectors, then show that is not an eigen (ce) Find the condition on k so that + 273+ is positive definite by stating the necessary result
   • (f) Identify the conic 5a? + + 5y? — 9 = 0. 5

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