Is my understanding of matrices correct?

Sometimes it is useful to write "matrices of vectors" that operate under the usual rules for matrix multiplication, as you try to do in your second-to-last equation. But this is very non-standard notation and I would avoid it if you're just starting out with linear algebra. As you say, you can represent $T$ by the image $T_J$ and $T_K$, in which case the representation of $T$ in coordinates is as a collection of column vectors: $$T = \left[\begin{array}{c|c} & \\ T_J & T_K \\ & \end{array}\right]$$ With $Tv$ taking linear combinations of the two columns.

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Linear Algebra

Matrices

Intuition

Linear Transformations

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