PART 13

4 a) Define kernel ofa linear transformation. Show that kernel of a linear
transformationT: V Wis a subspace of V. (4 marks)

b) Show that the function T from R > intoR 2 defined by
T ( X1,X2, .13) ಎ ( X1 + x2 , 2x3 — (1) is a linear transformation. Find the matrix of
T relative to the standard bases of R 3 andR 2,(5 marks)

5 a) Find an orthonomal basis for the subspace spanned by the vectors
( 1,0,1)4 in R (4 marks)

b) State and prove rank nullity theorem. (5 marks)

6 a) Let 8 = (bl, b2)andC (വ, be bases for R 2 where bl = (7,5), ),൨ =( 1,
—5) and % ಎ (— ). Find the transition matrix fromB toc and from (10 8. (4
marks)

b) State andprove Cauchy - Schwarz inequality. (5 marks)

PART C
2 2 1
7 a) Find the eigen values of A =3 I 5 Also find a basis for the 2
eigen space of eacheigen value. (6 marks)
1 4]
-2 2
b)Find the singular value decomposition of A = 2 -2./2 (6 marks)
-1 4 -2
8 a) If possible, diagonalize = |-3 —3 4
-3 0 (6
marks) 3

0) Define a Hermitian matrix. Prove that the eigen values of a Hermitian matrix are
real. (6 marks)
9 a) Construct a spectral decomposition of A = (6 marks)

b) Examine the definiteness of the quadratic form Q(x) = 3X1 + 2X2 + X3 + 4x1x2
+ 4x2x3(6 marks)