BSc Mathematics SEM V 2016 17 2016-17 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. If > R” is such that 8
   • (ii) — = — for all x,y then, show that T is an orthogonal linear transformation
   • ii. State and prove the Cayley Hamilton theorem. 8
   • (b) Answer any TWO
   • i. Show that any orthogonal linear transformation R? — R? is either a rotation about origin or a reflection line passing through origin 6
   • ii. Let V be a finite dimensional inner product vector space. Let 7’: V + V be a linear transformation. Prove is orthogonal if and only if = 6
   • iii. Show that T : R? defined by = is an isometry. Express it as a composition of an orthogonal transformation and a translation map 6
   • iv. Let A= |2 Find the characteristic polynomial of A and using 6
2. Q2 (a) Answer any ONE
   • i. Show that a real square matrix with nm real eigen values is similar to an upper triangular matrix 8
   • ii. Define minimal polynomial of a square matrix. Show that the minimal poly- nomial of a real square matrix A divides every polynomial that annihilates (Polynomial with real coefficients f(x) annihilates A if f(A) =0 ) 8
   • (b) Answer any TWO
   • i. Define eigen value of a real square matrix. Prove that \ R is an eigen value of if and only if a root the characteristic polynomial 6
   • ii. Show that eigen vectors v1, , corresponding to distinct eigen values respectively of a square matrix A are linearly independent 6
   • iii. Let V be a vector space of dimension 3 and v2, v3} be a basis of V. Find eigen values and corresponding eigen spaces of V — V be defined by 6
   • iv. Let be a real matrix. if A has n distinct characteristic roots, then prove that the characteristic polynomial of A = the minimal polynomial of 6
3. Q3 (a) Answer any ONE
   • i. Show that a matrix is orthogonally diagonalizable if and only if there is an orthonormal basis of R” consisting of eigen vectors of A 8
   • ii. Show that if is eigen value of real symmetric matrix A, then \ R. Also prove that the eigen vectors associated with distinct eigen values of A are 8
   • (b) Answer any TWO
   • i. Show that algebraic multiplicity of an eigen value of a square matrix is greater than or equal to its geometric multiplicity ii, Let A = Find a non-singular matrix P such that is a (6) diagonal matrix and hence find 6
   • iii. Let A is diagonalizable matrix such that eigen values of A are 6
   • (p) +1, then show that A is invertible and A =
   • (q) 0 and 1 then show that A? = A
   • iv. Show that every quadratic form over R can be to reduced to standard form by-an orthogonal change of variables X = = y= and P is an n x n orthogonal matrix 6
4. Q4 Answer any THREE
   • (a) Find an orthogonal transformation in R? which represents reflection with respect to the plane x 5
   • (b) Let V = and W = Space of 2 x 2 real symmetric matrices. Find a basis of W and the quotient space V/W 5
   • (c) Let be an orthogonal matrix with det A = —1, then show that -1 is an eigen value of .A 5
   • (d) Find the minimal polynomial of the diagonal matrix A = Show that A= , a,b, d R is diagonalizable if and only if b =0 ora d. (5) 5
   • (f) Let A= G ay a Ris a parameter, then show that A is diagonalizable if 5

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