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 Fill in the blank by choosing the correct. option
   • i. Let V = = +23= 0},
   • (a) (b) (c) (d) None of these
   • ii. is ——— 2
   • (a) an orthogonal matrix of reflection
   • (b) an orthogonal matrix of rotation
   • (c) not an orthogonal matrix
   • (d) None of these
   • iii. Let A= then = ———- 2
   • (a) (b) (c) None of these
   • iv. If characteristic polynomial of A is + and characteristic polynomial of is + byt +
   • (a) 1 (d) None of these
   • v. If 2 is an eigen value of a linear transformation T:R” > then one of the eigen values of T? — is ———
   • (a) (c) (d) None of these
   • vi. Let be a real matrix of rank 1, then the eigen values of A are
   • (c) 0 and det (d) None of these
   • vii. The minimal polynomial of the diagonal matrix 2
   • (a) (c) None of these
   • viii. If non-zero, non-diagonal A, B such that A? = J, 2
   • (a) only A is diagonalisable. (b) only B is diagonalisable
   • (c) both A and B are diagonalisable. (d) None of these
   • ix. A= 2)
   • (a) diagonalisable but not orthogonally diagonalisable
   • (b) orthogonally diagonalisable
   • (c) not diagonalisable
   • (d) None of these
   • x. Rank and signature of the quadratic from Q(x) =
   • (c) None of these
2. Q2 (a) Attempt any ONE
   • i. Let V be a finite dimensional inner product vector space and T:V—V bea 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 also an orthonormal basis of V
   • ii. State and prove the ’First Isomorphism Theorem of vector space’ (Fundamental theorem of vector space homomorphism)
   • (b) Attempt any TWO
   • i. Show that any orthogonal linear transformation R? is either a rotation about origin or a reflection about a line
   • ii. Let be a real matrix. If ag is the constant term of the polynomial det — A) then show that a, = det A 6
   • iii. be a diagonal matrix over R with characteristic
   • iv. Let A be n x n real matrix. Express the characteristic poly- nomial of A in terms of the characteristic polynomial of A where a R. Hence or otherwise show that, if is nilpotent then the characteristic polynomial of A — J, is 6
3. Q3 (a) Attempt any ONE
   • i. Let be a real matrix and be the distinct eigenvalues of A with as corresponding eigen vectors, then show that X2,--- ,X, are linearly indepen 8
   • ii. Define minimal polynomial of a suare matrix. Show that a root of the minimal polynomial of matrix A if and only if a is a characteristic root of A 8
   • (b) Attempt any TWO
   • i. Define invariant subspace. Let V be a finite dimension vector space and V + V be a linear transformation. Show that ker [mT are invariant under T’
   • ii. Let A and B matrices. Prove that if Aand Bare similar then characteristic polynomial of A = characteristic polynomial of B. Is the converse true? Justify 6
   • iii. Let be an eigen value of n x n matrix A. Show that any non-zero column of adj (A is an eigen vector of A cor responding to Ao
   • iv. Find the characteristic polynomial and the minimal polyno
4. Q4 (a) Attempt any ONE
   • 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
   • ii. Show that any real symmetric matrix is orthogonally diago- 8
   • (b) Attempt any TWO
   • i. Show that every quadratic form over R can be reduced to standard form by an orthogonal change of the variables X = PY, X = Yn)! and P is ann x n orthogonal matrix 6
   • ii. Show that eigen vectors associated to distinct eigen values of a real symmetric matrix are orthogonal
   • iii. Show that A = is diagonalizable if and only if b = 0 or a # d. [Turn over] 6
   • iv. Let A be a square matrix of order n such that A. Show that A is diagonalizable 6
5. Q5 Attempt any THREE
   • (a) If A= 3) Find — A? —4A — using the Cayley
   • (b) Let A and B ben x n real matrices. If AB are orthogonal matrices then prove that B and BA are both orthogonal matrices
   • (c) Find the eigenvalues and the bases of the corresponding eigen
   • (d) If is the characteristic polynomial of a matrix then find the characteristic polynomial of (i) (ii) (iii) A? Show that A= is diagonalisable. (5)
   • (f) Identify the conic 2x7 — — y? + 8. 5

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