10.

11.

12.

13.

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16.

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18.

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2 C 21545

If T, :R” > R” is a matrix transformation. Then define its kernel ker (T,) and Range of (ച).

What is ker (T,) in terms of null-space of A.

k, 0
Discuss the geometric effect on the unit square of multiplication by a diagonal matrix A= | 0 hy ۱

Confirm by multiplication that x is an eigen vector of A and find the corresponding eigen value, if
5 -1 1
A= =|. |
۲
Let R? have the weighted Euclidean inner product < u,v > = 2uyvy + 8ugvg. For u =(1,1),v =(3,2),
compute d(u,v).
If ॥ and v are orthogonal vectors in a real inner product space, then show that
2 2 2
|0| =| ५ || +| ४ |
State four properties of orthogonal matrices.

(10 x 3 = 30 marks)
Section B (Paragraph/ Problem Type Questions)

Answer at least five questions.

Each question carries 6 marks.

All questions can be attended.
Overall Ceiling 30.

Suppose that the augmented matrix for a linear system has been reduced to the row echelon form

solve the system.

-1 T
If Ais an invertible matrix, then show that A’ is also invertible and (க?) = (At) .

Let V be a vector space and 7, a vector in V and & a scalar. Then show that (i) OZ=0 ;

(1) (-1) ६ 5 -छ .

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