How to show if two matrix have linearly dependent columns

Question

I have a matrix $M \in \mathbb{R}^{n \times k}$, with rank $k$ and $n>k$ and a second matrix $L\in \mathbb{R}^{n \times l}$, with $l

If I want to show that the columns of $L$ are a linear combination of those of $M$, is that enough to show that

$$ L = MT $$

for a certain matrix $T\in \mathbb{R}^{k \times l}$ with full rank $l$?

Answer 1

Yes. For the first column of $L$ is the result of combining the columns of $M$ using the coefficients in the first column of $T$, and similarly for other columns.