Vector subspace and linear transformation

[Montgomery]

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.

My attempt:
1. (1, 1, 0, 3) and (1, 2, 1, 6)
2. Dim X = 2
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

 

[Mark44]

Please start a new thread in the Homework & Coursework section (Calculus & Beyond subsection). Homework-type problems have to be posted there, using the homework template. See the Physics Forums rules here: https://www.physicsforums.com/threads/physics-forums-global-guidelines.414380/, listed under Homework Guidelines.

When you start a new thread, please show more of your work, such as how you got the vectors of part 1, how you arrived at a dimension of 2 in part 2, and so on.