Linear algebra problem related to vector subspace

[Montgomery]

Homework Statement

X ={(x1,x2,x2 −x1,3x2):x1,x2 ∈R}
f(x1,x2,x2 −x1,3x2)=(x1,x1,0,3x1)
1. Find a basis for X.
2. Find dim X.
3. Find ker f and I am f
4. Find bases for ker f and I am f
5. Is f a bijection? Why?
6. Find a diagonal matrix for f.

Homework Equations

The Attempt at a Solution

1. Put x1=x2=1: (1, 1, 0, 3) and Put x1=1 and x2=2: (1, 2, 1, 6)
2. Dim X = 2 as there are two vectors
3. Ker f = 0, I am f = X
4. (0,0,0,0)
5. I guess no, but do not know how to explain
6. No idea

 

[Fredrik]

1. Looks like a good guess, but you will need to prove that the two vectors you found span X.

2. Here you will also need to prove that the two vectors you found in 1 are linearly independent.

3. ker f is a set, not a vector or a number. If you meant ker f={0}, where 0 is the zero vector, then the answer is wrong since e.g. f(0,1,1,3)=0. Your answer for I am f is wrong too, since (1,2,1,6) is in X but not in I am f.