Friedberg Linear Algebra problems

Question

In Friedberg's Linear Algebra 4th edition, I have trouble understanding 2 related questions:

Section 2.1 problem 34: "Let $V$ and $W$ be vector spaces over a common field, and let $\beta$ be a basis for $V$. Then for any function $f: \beta \to W$ there exists exactly one linear transformation $T: V \to W$ such that $T(x) = f(x)$ for all $x \in \beta$.

I don't understand what the restriction of $T$ to the basis space $\beta$ does at all, and how does one prove that there is "exactly one" transformation? Directly related to this,

Section 2.7 problem 18: "Let $V$ be a nonzero vector space over a field $F$ and let $S$ be a basis for $V$. Let $\Phi: V^* \to \mathcal{L}(S,F)$ be the mapping defined by $\Phi(f)=f_S$, the restriction of $f$ to $S$. Prove that $\Phi$ is an isomorphism. Hint: Apply Exercise 34 of Section 2.1. (the above problem)

I don't see how the two problems are related except for the fact that there is a linear transformation and there is a linear transformation restricted to a subset containing only basis vectors.

Answer 1

For the first exercise, this is simply the fact that a linear transformation is completely determined by its effect on a basis. One proves that there is exactly one by using linearity.

The second problem is a direct application of that fact. It basically says that dual space is isomorphic to the space of linear functions on $V$ into $\mathcal F$. These are also known as linear functionals. Just let $W=\mathcal F$, the one dimensional vector space over $\mathcal F$, in the first part.

It might have been better if Freidberg had written $\mathcal L(\color{blue}{V},\mathcal F)$, in the second question, I think. Since a basis is not a vector space.

Answer 2

First, there is a typo in the statement of the second problem: it should be $\mathcal F(S,F)$ instead of $\mathcal L(S,F)$. Recall that, following the notation of the book, $\mathcal F(S,F)$ is the set of all functions from $S$ to $F$; naturally $\mathcal F(S,F)$ is a vector space over $F$ with the addition and the scalar multiplication defined pointwise.

Now, once observed this, if you put $W = F$ and $\beta=S$ in the first problem, this one says that:

If $V$ is a vector space over a field $F$ and $S$ is a basis for $V$, then for any function $f : S \to F$ there is a unique linear map $\tilde{\vphantom{}f} : V \to F$ such that $\tilde{\vphantom{}f}|_S = f$, in other words, any function in $\mathcal F(S,F)$ can be extended to a function in $V^*$ in a unique way.

Thus, if $\Phi : V^* \to \mathcal F(S,F)$ is the function given by $\Phi(g) = g|_S$, the statement written above means precisely that $\Phi$ is a bijection (the existence part means that $\Phi$ is a surjection and the uniqueness part means that $\Phi$ is an injection). Therefore, it only rest to prove that $\Phi$ is linear.