BSc Mathematics SEM V 2019 20 Oct 2019-20 MATHS MATHEMATICS LINEAR ALGEBRA 11.10.19 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) Let A= 0030 = M,(R) : AB = BA}. Then, 2
   • (a) dim V =4 and dim M,(R)/V = 12
   • (b) dim V = 8 and dim M,(R)/V =8
   • (c) dim V and dim M,(R)/V = 16
   • (d) None of these
   • (ii) Let a@ be an orthogonal transformation of the plane such that the matrix of a w. r. t. the standard basis of R? is ( v2 v2
   • (c) rotation about the line y= None of the above
   • (iii) A 2 x 2 matrix A has the characteristic polynomial then the value of det + A) is
   • (a) 2 (d) 2de
   • (iv) Let A and B be square matrices such that AB then 2
   • (a) A but not (b) B but not A
   • (v) If A is a characteristic root of a matrix A then characteristic roots of —A and al — A respectively are 2
   • (a) —A anda
   • (c) —A and (d) None of these
   • (vi) Which of the following statements are true 2
   • (p) If the characteristic roots of two n x n matrices are same then their characteristic polynomials are same
   • (q) If the characteristic polynomials of two n x n matrices are same then their characteristic roots are same
   • (r) If eigen values of two n x n matrices are same then their eigen vectors are same
   • (s) The characteristic roots of two n x n matrices are same but their characteristic polynomials may not be same
   • (a) (q) and (s) are true. (b) (p), (r) are true
   • (c) (p), (q) and (r) are only (q) is true
   • (vii) The minimal polynomial of the diagonal matrix 2
   • (a) (c) (d) None of these
   • (viii) Let A= 2
   • (a) A is orthogonally diagonalizable if and only if a = 1
   • (b) A is not diagonalizable for any a R
   • (c) A is diagonalizable but not orthogonally diagonalizable
   • (d) None of these
   • (ix) If A,B,C,D M2(R) such that A,B,C,D are non-zero and not diagonal. If A? = [, B? = B,C? =0,C 4 0 and every eigenvalue of D is 2, then 2
   • (a) A,B,C, D are all diagonalizable
   • (b) B,C, D are diagonalizable
   • (c) A, B are diagonalizable
   • (d) Only D is diagonalizable
   • (x) The quadratic form Q(x) = + + has 2
   • (c) rank = 2, signature = 2. (d) None of the above
2. Q2 (a) Answer any ONE
   • (i) Let V be a finite dimensional inner product vector space and T:V—V bea linear transformation. Prove that the follow ing statements are equivalent
   • (p) T is orthogonal
   • (r) If is an orthonormal basis of V, then {T(e;) is orthonormal basis of V
   • (ii) State and prove the Cayley Hamilton Theorem. 8
   • (b) Answer any TWO
   • (i) State and prove the ’First Isomorphism Theorem of vector space’ (Fundamental theorem of vector space homomorphism) 6
   • (ii) Let (V,<>) be an n dimensional inner product space and W be a subspace of V of dimension n — 1. Let wu be a unit vector orthogonal to W. Show that V V defined by T(x) = x — is an orthogonal linear transformation such that T(w) = w, Vw W and T(u) defined by T(x) = AX(X being a column vector in Find kerT,a basis of kerT and
   • (iv) Show that a : > defined by = + v3 — ly, — is an isometry. Express it as a composite of an orthogonal transformation and a translation
3. Q3 (a) Answer any ONE
   • (i) Define eigen value of a real square matrix. Show that, if an eigen value of a real n x n matrix A, then
   • (p) A is an eigen value of A’
   • (q) A* 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) Show that minimal polynomial of a real matrix divides every polynomial which annihilates A. Further show that is a root of the minimal polynomial of matrix A if and only is a characteristic root of A 8
   • (b) Answer any TWO
   • (i) matrix, and , A, are distinct eigen value of A with as corresponding eigenvectors, then show that are linearly independent
   • (ii) Let Anxn be a real matrix. if A has n distinct characteristic roots, then prove that the characteristic polynomial of A = the minimal polynomial of A
   • (iii) Find the eigen values and eigen vectors of such that 6
   • (iv) Let be a non-zero vector in and A = vv? where v is treated as column vector. Find the minimal polynomial
4. Q4 (a) Answer any ONE
   • (i) Define an orthogonally diagonalizable matrix. Show that ev- ery real symmetric matrix is orthogonally diagonalizable 8
   • (ii) Let A be real symmetric matrix of order n. Show that teristic roots of A are real. Also show that if A are distinct eigen values of A and are corresponding eigen vectors then X,, X2 are orthogonal 8
   • (b) Answer any TWO
   • (i) Show that an n x n matrix A is diagonalizable if and only if IR" has a basis consisting of eigen vectors of A 6
   • (ii) Show that every quadratic form over reduced to standard form by an orthogonal change of variables X = PY ,where X,Y are column vectors in R and P is an n X n orthogonal matrix 6
   • (iii) For A = find a non-singular matrix P such that is a diagonal matrix. Hence or otherwise find and B such that B? = A
   • (iv) Let A be a 13 x 13 real matrix of rank 1. Find the eigen val ues and show that geometric multiplicity is equal to algebraic multiplicity for each eigen value
5. Q5 Answer any FOUR
   • (a) Find the orthogonal transformations in which represent reflec tions with respect to 2x —y+
   • (b) Express the characteristic polynomial of aJ + bA interms of the characteristic polynomial of A
   • (c) Prove or matrices are similar if and only if their characteristic polynomials are same
   • (d) Define invariant subspace with respect to a linear transformation Check which of V = : R} andW = : R} are R? + R? invariant where = (y,x). Also identify the eigen value of T’ with respect to which W is an eigen space
   • (e) If A? = A for non-zero n n matrix A then show that algebraic multiplicity of eigen value 1 is rank A
   • (f) Find value of k, for which the symmetric matrix associated to the quadratic form is positive definite State the result used

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