Exercise 1.6.21

Sufficiency: If the vector space $V$ is finite-dimensional, say dim$=n$, and it contains an infinite linearly independent subset $\beta$, then we can pick an independent subset ${\beta }^{\prime }$ of $\beta$ such that the size of ${\beta }^{\prime }$ is $n+1$. Pick a basis $\alpha$ with size $n$. Since $\alpha$ is a basis, it can generate $V$. By Replacement Theorem we have $n\ge n+1$. It’s a contradiction. Necessity: To find the infinite linearly independent subset, we can let $S$ be the infinite-dimensional vector space and do the process in exercise 1.6.20(a). It cannot terminate at any $k$ otherwise we find a linearly independent set generating the space and hence we find a finite basis.