Exercise 3.D.20

Proof. Let $v_k=\left(A_{1,k},\dots ,A_{n,k}\right)$, for $k=1,\dots ,n$.

Suppose (a) holds. Then (a) implies that $v_1,\dots ,v_n$ is linearly independent. Since this list has length $\dim <!--\; FUNCTION\; APPLICATION\; -->{𝔽}^n$, it must also span ${𝔽}^n$, proving (b).

Suppose (b) holds. Then (b) implies that $v_1,\dots ,v_n$ spans ${𝔽}^n$. Moreover, this list has length $\dim <!--\; FUNCTION\; APPLICATION\; -->{𝔽}^n$, therefore it is linearly independent, proving (a). □