# Prove that set of linear transformations is infinite-dimensional if domain of linear maps is finite-dimensional and codomain is infinite-dimensional

Suppose $$V$$ is finite-dimensional with $$dim\ V > 0$$, and suppose $$W$$ is infinite-dimensional. Prove that $$\mathcal{L}(V,W)$$ is infinite-dimensional.

Notation: V, W are vector spaces over complex or real numbers. L(V,W) stands for set of linear maps/linear transformations from V to W.

My question is regarding this solution(source:https://linearalgebras.com/3a.html): why is it correct to look at the particular value of the functions $$T_1,...,T_m$$ on $$v=v_1$$ to prove that the list of functions(in particular linear transformations) $$T_1,...,T_m$$ is linearly independent? Shouldn't the implication of the statement:

$$a_1T_1(v_1)+a_2T_2(v_1)...+a_mT_m(v_1)=0$$ implies $$a_1=...=a_m=0$$

be

"$$T_1(v_1),T_2(v_1),...,T_m(v_1)$$ is a list of elements of $$W$$ that is linearly independent."

"$$T_1,T_2,...,T_m$$ is a list of elements of $$\mathcal{L}(V,M)$$ that is linearly independent."?
Why isn't this proof as erroneous as saying that "the list of polynomials(reals to reals) $$f,g$$ in which $$f(x)=x$$ and $$g(x)=x^2$$ is linearly dependent because $$a_1f(0)+a_2g(0)=0a_1+0a_2=0$$ is true for multiple real values of $$a_1,a_2$$"?
• $T_1,\dots,T_m$ linearly dependent means there exist $a_1,\dots,a_m$ not all zero such that $a_1T_1+\cdots+a_mT_m$ is identically zero, that is, such that for all $v$, $a_1T_1(v)+\cdots+a_mT_m(v)=0$. But we know there are no such $a_i$ for $v=v_1$, hence, the maps are linearly independent. Commented Jan 26, 2022 at 23:06
So the fact that $$T_1(v_1), T_2(v_2), ...$$ are linearly independent is trivial because they were defined to be that way. The implication is that we assumed $$a_1T_1 + a_2T_2 + ... + a_mT_m = 0$$ and now we have shown that $$a_1, ..., a_m = 0$$. Notice that there are no $$v_i$$'s in that line. Thats because this is a linear combination of linear maps, not vectors. So we have shown this linear combination of functions is the 0 function.