Exercise 3.A.12

Alternate Proof. Recall from Exercise 14 in Section 2.A that if there exists a sequence of vectors $v_1,v_2,\dots$ in $V$ such that $v_1,\dots ,v_m$ is linearly independent for every positive integer $m$, then $V$ is infinite-dimensional. The converse is also true.

In this exercise, $W$ is infinite dimensional, implying there exist vectors $w_1,w_2,\dots$ in $W$ such that $w_1,\dots ,w_m$ is linearly independent for every positive integer $m$. We wish to show that there exists a sequence of linear mappings $T_1,T_2,\dots \in L(V,W)$ such that $T_1,\dots ,T_m$ is linearly independent for every positive integer $m$.

Since $V$ is finite-dimensional, it has a basis, say $v_1,\dots ,v_n$. Then, by proposition 3.5 on page 54, we can define unique linear mappings $T_i{v}_1=w_i$ for $i=1,2,\dots$. Clearly, $T_i\in L(V,W)$ implying the existence of a sequence $T_1,T_2,\dots \in L(V,W)$ where $T_1{v}_1=w_1,T_2{v}_1=w_2,\dots$.

Suppose there exist $a_1,\dots ,a_m\in 𝔽$ for any $m\in Z^{\text{+}}$ such that $a_1{T}_1+\cdots +a_m{T}_m=0$. Then,

implies

implying

Since $w_1,\dots ,w_m$ are linearly independent, it implies $a_1=\cdots =a_m=0$; by the equation $(a_1{T}_1+\cdots +a_m{T}_m)v_1=0$, it further implies $T_1,\dots ,T_m$ are linearly independent. Hence, $L(V,W)$ is infinite-dimensional. □