Kernel and image of a matrix converting linear transformation

Question

Let $V, W$ be finite-dimensional vector spaces, and let $\alpha$ and $\beta$ be fixed bases for $V$ and $W$ respectively.

Define the transformation $\Omega: L(V,W) \to M_{m \times n} (\mathbb{R})$ where $L(V,W)$ is the set of all linear transformation where its domain is $V$ and codomain is $W$ and $M_{m \times n} (\mathbb{R})$ is the set of all $m \times n$ matrices with real number entries. $\dim(V) = n$ and $\dim(W) = m$.

$\Omega(T)= [T]^\beta _\alpha$ for any $T$ where it $T$ is a linear transformation.

$[T]^\beta _\alpha$ is the matrix with respect to $\beta$.

(So something like taking the vectors in $\alpha$ and applying the linear transformation $T$ on the vectors in $\alpha$ then writing it as a linear combination of the vectors in $\beta$ and using the coefficients of the linear combination which becomes the column vectors... And do this for all the vectors in $\alpha$)

I really hope that I used the notation properly for the support of help with the problem I have.

Question:

(1) What is the kernel of the linear transformation $\Omega$?

(2) What is the image of the linear transformation $\Omega$?

And I am having trouble answering both of the questions above.

Attempted answer:

By definition of a kernel of linear transformation $\Omega$ is:

$\ker(\Omega)=\{T\in L(V,W) : \Omega(T)=0\}$ where $0$ represents the zero matrix where all entries are all $0 \in \mathbb{R}$.

By how $\Omega(T)$ is defined we can rewrite $\Omega(T)=[T]^\beta_\alpha=0.$

So I am confused as to how I am supposed to find the elements for the kernel of the linear transformation... Not sure how I can further continue from what I have above.

Secondly, the image of the linear transformation I have no clue on.

Any help, tips would be appreciated! Thank you in advance.

Answer 1

For the first part you're almost there, you have proven that some transformation $T$ from $V$ to $W$ that has matrix $[T]^\beta _\alpha$ is in the kernel if $[T]^\beta _\alpha =0$. Now to conclude can a transformation $T \not \equiv 0$ have matrix $[T]^\beta _\alpha=0$?

For the other part I'd suggest the following:

Hint: Consider the matrices $E_{i,j}$ that have $1$ in row $i$ and column $j$ and $0$ elsewhere. You could prove this matrices are a basis of $M_{m \times n} (\mathbb{R})$ and from here try to find a transformation for each one of them.