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1 -2 0 0 3
2-5 -3 2 6

26. (a) IfAisthematrix}0 5 15 10 0|,thenfindabasisfortherow space consisting on entirely
2 6 18 8 6

row vectors from A.

(b) Find the standard matrix for the operator T: R® ‏ع‎ R® that first rotates a vector counter

clockwise about z-axis through an angle 0, reflects the resulting vector about yz plane and

then projects that vector orthogonally onto the xy plane.

1
27. (a) OnP,, polynomial in [-1,1], define innerproduct as < P,¢ >= J p(~)a(x) dx . Find | 7 |, | 9 |
1

and for p =x andq =x.

(b) IfAisann xn matrix with real entries, show that A is orthogonally diagonalizable if and only
if A has an orthonormal set of n eigenvectors.
(2 x 10 = 20 marks)

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