Whats the answer for these linear transformations questions?

Question

Q1: use function to find (a) the image of v and (b) the preimage of w.

1-T(v₁,v₂)=(v₁+v₂,v₁-v₂),v=(3,-14),w=(3,19)

2-T(v₁,v₂,v₃)=(v₂-v₁,v₁+v₂,2v₁),v=(2,3,0),w=(-11,-1,10)

Q2:determine whether the fuction is a linear transformation(explain).

1-T:R²→R²,T(x,y)=(x,1)

2-T:R³→R³,T(x,y,z)=(x+y,x-y,z)

Q3:let T:R³→R³ be a linear transformation such that T(1,0,0)=(2,4,-1),T(0,1,0)=(1,3,-2),and T(0,0,1)=(0,-2,2) Find :

23-T(0,3,-1)

Q4:the linear transformation T:Rⁿ→R^m is defined by T(v)=Av.Find the dimensions of Rⁿ and R^m.

1-A= ⌈ 0 1 -2 1⌉

-1 4 5 0

⌊ 0 1 3 1⌋.

2-A= ⌈ 1 2 ⌉

-2 4

⌊-2 2 ⌋.

Answer 1

4) 1- matrix A have order 3*4 then v have order 4*1

then T(v) have 3*1

mean n=4 and m =3