Exercise 2.A.14

Proof. Suppose $V$ is infinite-dimensional. Choose any positive integer $m$ and consider the following process

• Step 1: Choose a non-zero vector $v_1$ in $V$.
• Step j: Because $V$ is infinite-dimensional, it follows that $v_1,\dots ,v_{j-1}$ doesn’t span $V$. Hence, there is a vector $v_j\in V$ such that $v_j\notin \mathrm{span}<!--\; FUNCTION\; APPLICATION\; -->(v_1,\dots <!--\; FUNCTION\; APPLICATION\; -->,v_{j-1})$. Therefore $v_1,\dots ,v_j$ is a linearly independent list in $V$. If $j=m$ stop the process.

After step $m$, the process stops and we have constructed a linearly independent list of length $m$.

For the converse, we will prove the contrapositive. Suppose that $V$ is finite-dimensional and $v_1,\dots ,v_n$ spans $V$. By 2.23, we cannot have a linearly independent list of arbitrary length (specifically, it cannot be greater than $n$).

Therefore, by modus tollens, if we can have a linearly independent list in $V$ of arbitrary length, then $V$ is infinite-dimensional. □