9. (i) Define linear transformation from a vector space Vto a vector space Wover the

same field F. Define the null space and image of 0. Prove that the null space is a

subspace of V and the image is a subspace of W.

(ii) Determine which of the following are linear transformations from R³ to itself, and

for those which are, find the null space and image.

(a) 0₁

(b) 0₂

I

(-)-(

=

(c) 03

(d) 0₁

x

Yy

x

Y

2

=

=

().

=

2x + y + 1

y-z-2

2+x+5

x²

xy

x+y+z

y-z+x

z + 2x + 3y

2x + 3y + 4z

x-y + 2z

x +9y2z