$T$, $S$ are lineary dependent $\Leftrightarrow$ $[T]_B$, $[S]_B$ are lineary dependent.

Question

EDIT: I see in the comments that my question is not clear enough so I will explain:

if I want to check whether $T$, $S$ are linearly independent or not, I will just pick an easy to work with, basis for $V$ and then check if $[T]_B$, $[S]_B$ are linearly independent. so the real question is can I do that? and why?

Answer 1

Every vector space $V$, such that dim$(V)=n$ is isomorphic to $\mathbb{F}_n$ (column vectors with $n$ components), but even more than this, the isomorphism is a linear isomorphism, that is it preserves the structure of the vector space. To prove that this is indeed so you have to prove that the function $\phi$ mapping each vector of $V$ to its coordinate vector, is:

• functional
• injective and surjective
• preserves scalar multiplication and vector addition (this is where you add the extra conditions of preserving the structure of the vector space)

It is a quite straightforward direct proof, and most linear algebra books have it.

I hope this helps.