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3 C 21545

If S= [ण,०७, ‏بے‎ Un | is a basis for a vector space V, then show that every vector v in V can be

expressed in form v =c vl; + Cg’9 +...+C,U, in exactly one way. What are the co-ordinates of V

relative to the basis S ?

Consider the basis B=[u,u,| and 8 =| ५, | for R?, where ५] = (2,2) Ug = (4,-1)
५1 = (1,8) > = (-1,-1).

(a) Find the transition matrix from B' to B.

(b) Find the transition matrix from B to 8

If A is a matrix with n columns, then define rank A, nullity of A and establish a relationship
between them.
Define eigen space corresponding to an eigen value , of a square matrix A. Also find eigen value
and bases for the eigen space of the matrix A = 3
Use the Gram-Schmidt process for an orthonormal basis corresponding to the basis vectors
uy = (1,11), ug =(0,1,1) and ug =(0,0,1).

(5 x 6 = 30 marks)

Section C (Essay Type Questions)

Answer any two questions.
Each question carries 10 marks.

Show that the following statements are equivalent for an n x n matrix A:
(a) Ais invertible.
(b) Ax =0 has only the trivial solution.
(c) The reduced row echelon form of A is I,,.

(d) A is expressible as a product of elementary matrices.
(a) Define Wronskian of the functions f; = f,(x),fo =fo(x)... fn =f, (x) which are n - 1 times
differentiable in (—s0, 0). Use this toshow that /; =x and f, =sinx are linearly independent

vectors in c™ (—, 0).

(b) Show that the vectors v, = (1,2,1),vg =(2,9,0) and v3 (3,3,4) form a basis for R®.

Turn over

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