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.