Whats the answer for these karnel and range of a linear transformation questions?

Question

Q1:find the kernel of the linear transformation.

1-T:R³→R³,T(x,y,z)=(0,0,0)

2-T:R³→R³,T(x,y,z)=(x,0,z)

3-T:R³→R³,T(x,y,z)=(z,y,x)

4-T:R²→R²,T(x,y)=(x-y,y-x)

Q2:the linear transformation T is represented by T(v)=Av.Find a basis for (a) the kernel of T and (b) the range of T.

1-A= ⌈ 1 2⌉

⌊ 3 4 ⌋

Q3:the linear transformation T is defined by T(x)=Ax.Find (a) ker(T), (b) nullity(T), (c) range(T), (d) rank(T).

1-A= ⌈ 4 1 ⌉

0 0

⌊ 2 -3⌋

Answer 1

R^3 is the kernel

(0, a, 0) is the kernel

(0, 0, 0) is the kernel

(a, a) is the kernel