Finding Kernel, Image, Rank, Nullity of Matrix

[Marakh]

Homework Statement

Find Kernel, Image, Rank and Nullity of the matrix
1 −1 1 1 
| 1 2 −1 1 |
0 3 -2 0 

Homework Equations

The Attempt at a Solution

I have reduced the matrix into rref of
3 0 1 3
0 3-2 0
0 0 0 0
But am struggling to find the column vectors which are the solutions of that, which I believe is the kernel
And then I know that the rank and nullity are the dimensions of the image and kernel respectively

 

[HallsofIvy]

The "kernel" of matrix A is the set of all vectors v such that Av= 0. If you write vector v as (a, b, c, d) then any vector in the kernel must satisfy a- b+ c+ d= 0, a+ 2b- c+ d= 0, and 3b- 2c= 0. The "image" is the set of all vectors, v, such that Ax= v for some vector x. If we write vector x as (a, b, c, d) then v must be (x, y, z) such that a- b+ c+ d= x, a+ 2b- c+ d= y, and 3b- 2c= z for some numbers a, b ,c, and d.