Question

Let V be a finite dimensional vector space and W a subspace of V. Prove that W is also finite dimensional.

Answer 1

Question 1. Let $V$ be a finite dimensional vector space and $W$ a subspace of $V$. Prove that $W$ is also finite dimensional.

Solution. Suppose the dimension of $W$ is infinite. This means that there is an infinite linearly independent subset $\{w_{i}\}_{i=1}^{\infty}$ of $W$. But $W$ is a subspace of $V$, in particular, $W \subseteq V$. So, $\{w_{i}\}_{i=1}^{\infty}$ is also a subset of $V$. Therefore, $\dim V = \infty$; a contradiction.