How to prove this vector basis claim?

Question

Let $B$ be a basis for $\Bbb R^n$. Prove that the vectors $v_1,v_2,...,v_k$ span $\Bbb R^n$ if and only if the vectors $[v_1]_B,[v_2]_B,...,[v_k]_B$ span $\Bbb R^n$.

I'm the beginner in mathematics? Could u help me pls.

Answer 1

We prove in general that if we have a surjective linear mapping $f: V \rightarrow W$ such that $V = span(A)$, then $W = span(f(A))$

In other words:

The image of a spanning set of $V$ is a spanning set for $W$ if the linear mapping is surjective.

Proof:

Let $A = \{\overrightarrow a_1, \dots \overrightarrow a_n\}$

Then,

$f(span(A)) = f(\{\sum \alpha_i \overrightarrow a_i\}) = \{\sum \alpha_i f(\overrightarrow a_i)\} = span(f(A))$

Let $V = span(A)$.

Then, by surjectivity of $f$, we have that $f(V) = W$.

Also: $f(V) = f(span(A)) = span(f(A))$

We deduce that $W = span(f(A))$ and this ends the proof.

Now, we know that the function $[o]_B: V \rightarrow \mathbb{R}^n: \overrightarrow v \mapsto [\overrightarrow v]_B$ is an isomorphism. Thus, that function is surjective. Applying what we proved above on this specific function, we find the desired result (notice that the if and only if follows because we can apply this result to the inverse isomorphism as well)