What do I use to find the image and kernel of a given matrix? I had a couple of questions about a matrix problem. What I'm given is:

Consider a linear transformation $T: \mathbb R^5 \to \mathbb R^4$ defined by $T( \vec{x} )=A\vec{x}$, where
  $$A = \left(\begin{array}{crc}
 1 &  2 &  2 & -5 &  6\\
-1 & -2 & -1 &  1 & -1\\
 4 &  8 &  5 & -8 &  9\\
 3 &  6 &  1 &  5 & -7
\end{array}\right)$$
  
  
*
  
*Find $\mathrm{im}(T)$
  
*Find $\ker(T)$

My questions are:
What do they mean by the transformation?
What do I use to actually find the image and kernel, and how do I do that?

I could give an explanation for the most appreciated answer why image is calculated in this way. Image of a matrix is basically all the vectors you can obtain after this linear transformation. Let's say $A$ is a $2 \times 2$ matrix
$$A=\pmatrix {a_1 & b_1\\ a_2 & b_2}$$
. If we apply A as a linear transformation to the standard base, aka the identity matrix, we get A itself. However, we could consider this transformation as it transforms the basis vectors to all the columns A has. (1, 0) to (a1, a2), (0, 1) to (b1, b2). Therefore, the image of A is just the span of the basis vectors after this linear transformation; in this case, span ((a1, a2), (b1, b2)). This is the reason why we need to get rref of the transpose of A. We are simplely getting linearly independent basis vectors after this linear transformation. If there's anything unclear, I really recommand you to watch this video made by 3Blue1Brown, it shows this in a visual way. Here's the link: https://www.youtube.com/watch?v=uQhTuRlWMxw
