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

Steps to Solve

1. Establish a Basis for V

Since $V$ is finite-dimensional and $\dim V > 0$, let's say $\dim V = n$ where $n$ is a positive integer. This means $V$ has a basis. Let ${v_1, v_2, ..., v_n}$ be a basis for $V$.

2. Use a Basis Vector and Define Linear Transformations

Consider one of the basis vectors of $V$, say $v_1$. Now, since $W$ is infinite-dimensional, we can find an infinite set of linearly independent vectors in $W$. Let ${w_1, w_2, w_3, ...}$ be a set of linearly independent vectors in $W$. For each $w_i$ in this set, define a linear transformation $T_i: V \to W$ as follows:

$T_i(v_1) = w_i$ $T_i(v_j) = 0$ for $j \neq 1$

Since ${v_1, v_2, ..., v_n}$ is a basis for $V$, a linear transformation from $V$ to any vector space is completely determined by its action on the basis vectors. Thus, each $T_i$ is a well-defined linear transformation.

3. Show Linear Independence of Transformations

We want to prove that the set of linear transformations ${T_1, T_2, T_3, ...}$ is linearly independent in $\mathcal{L}(V, W)$. To do this, consider a finite linear combination of these transformations that equals the zero transformation:

$$a_1T_1 + a_2T_2 + ... + a_kT_k = 0$$

where $a_i$ are scalars and $0$ denotes the zero transformation.

4. Apply the Linear Combination to a Basis Vector

Apply this linear combination to the basis vector $v_1$:

$$(a_1T_1 + a_2T_2 + ... + a_kT_k)(v_1) = 0(v_1) = 0$$

Using the linearity of the transformations, we have:

$$a_1T_1(v_1) + a_2T_2(v_1) + ... + a_kT_k(v_1) = 0$$

Substituting $T_i(v_1) = w_i$, we get:

$$a_1w_1 + a_2w_2 + ... + a_kw_k = 0$$

5. Conclude Linear Independence of Coefficients

Since $w_1, w_2, ..., w_k$ are linearly independent vectors in $W$, the only way for this linear combination to equal zero is if all the coefficients are zero:

$$a_1 = a_2 = ... = a_k = 0$$

This shows that the linear transformations $T_1, T_2, ..., T_k$ are linearly independent in $\mathcal{L}(V, W)$. Since we can find an infinite set of linearly independent transformations in $\mathcal{L}(V, W)$, it follows that $\mathcal{L}(V, W)$ is infinite-dimensional.