If a 3x3 matrix multiplied with a 3x3 matrix returns the transform of the second 3x3 matrix, what does a nonsquare matrix multiplied with 3x3 return? I have just recently been introduced to matrices. What Ive understood for example, is that a 3x3 matrix can be thought of as a vector, the first column representing the x component/ x axis basis, second column the y component/y axis basis and so on.
So a 3x3 matrix multiplied with a 3x3 matrix returns us a matrix where the vectors of the second 3x3 matrix have been transformed by the first 3x3 matrix.
This makes sense, the x-component of the first matrix is treated as basis and the x coordinate of each of the x y and z components of the second matrix is transformed as per the basis we got from the first matrix, rinse repeat for the rest of the second matrix
But for something like say  2x4 matrix A multiplied with a 2x2 matrix B, how can the same logic be applied in the same intuitive manner?
Matrix A is a vector containing 4 components in the x y z axis and whatever the 4th axis is, and each of these components have 2 coordinates x and y.
Matrix B on the other hand has 2 components X and Y, each with x and y coordinates for that specific component
When A transforms B, how would the Z component of A transform matrix B if B has no Z component at all? Similarly how does the 4th "dimension"'s component  transform B if it doesn't have that component?
Not to mention , the Z and 4th dimension's components themselves are only represented by 2 coordinates x and y. How can something like Z which is part of 3d be represented by just 2 vectors X and Y? Is it because 'Z' and the 4th dimension vector here are not actually in 3d or 4d as they are in real life but are just some vectors that are restricted to two dimensions like the x and y component?

I guess you are a beginner to the subject. Well, you can't think multiplication between matrices as the multiplication with numbers. You know you can multiply any real number with another real number for instance, but this doesn't hold true between matrices. You can multiply a $n\times p$ matrix  $A$ with a $m\times q$ matrix $B$ if and only the number of columns of $A$ is equal to the number of rows of $B$, in other words $p=m$. This happens because of the definition of multiplication between matrices.
You said $A$ is $2\times4$ and $B$ $2\times2$, so you can't write $AB$ as $4\neq2$, but you can compute $BA$ which will be now a $2\times4$ matrix.
