Is a Linear Map uniquely determined on $V$ or on a basis of $V$?

Question

In Linear Algebra Done Right by Axler, there are two sentences he uses to describe the uniqueness of Linear Maps (3.5) which I cannot reconcile. Namely, whether the uniqueness of Linear Maps is determined by the choice of 1) basis or 2) subspace. These two seem like very different statements to me given there can be a many-to-one relationship between basis and subspace. In otherwords, saying a Linear Map is "unique on a subspace" seems like a stronger statement than saying it is "unique on a basis".

This first sentence he writes before proving the theorem (3.5):

The uniqueness part of the next result means that a linear map is completely determined by its values on a basis.

This second sentence he writes at the end after proving the uniqueness of a linear map:

Thus $T$ is uniquely determined on $span(v_1, \dots, v_n)$ by the equation above. Because $v_1, \dots, v_n$ is a basis of $V$, this implies that $T$ is uniquely determined on $V$.

My question is, if $T$ is uniquely determined on $V$, doesn't that imply that the choice of basis for $V$ doesn't matter (since each basis of $V$ spans $V$)? But if so, a key part of the theorem requires explicitly choosing $T$ such that $T(v_j)=w_j$, meaning if we choose a different basis $a_1, \dots, a_n$ of $V$ and then select $T$ such that $T(a_j)=w_j$, we would get a different $T$.

I've found these questions here and here that are tangentially related but don't address my question specifically. On the other hand the questions here and here get a little closer, but the answer in the first suggests that the choice of basis is arbitrary where as the answer in the second suggests the basis must be the same.

For more information, I've included the complete statement of the theorem plus the last paragraph of the proof.

Theorem 3.5 Linear maps and basis of domain

Suppose $v_1, \dots, v_n$ is a basis of $V$ and $w_1, \dots, w_n \in W$. Then there exists a unique linear map $T: V \to W$ such that $Tv_j=w_j$ for each $j = 1, \dots, n$.

Last paragraph of proof:

To prove uniqueness, now suppose that $T \in \mathcal{L}(V,W)$; and that $Tv_j=w_j$ for each $j = 1, \dots ,n$. Let $c_1, \dots, c_n \in F$. The homogeneity of $T$ implies that $T(c_jv_j) = c_jw_j$ for each $j=1, \dots, n$. The additivity of $T$ now implies that $T(c_1v_1 + \cdots + c_nv_n) = c_1w_1 + \cdots + c_nw_n$. Thus $T$ is uniquely determined on $span(v_1, \dots, v_n)$ by the equation above. Because $v_1, \dots, v_n$ is a basis of $V$, this implies that $T$ is uniquely determined on $V$.

Answer 1

Not an answer, just too long for a comment.

Suppose I have a transformation $T: \mathbb{R}^2 \to \mathbb{R}$.

Pick a basis $B=\{ e_1=(1,0), e_2=(0,1) \}$.

Then $T$ is completely specified by the values $w_1=T(e_1),w_2=T(e_2)$.

Now suppose I pick another basis $C=\{ c_1=(1,1), c_2=(1,-1) \}$.

If I pick values $v_1,v_2$ corresponding to the points $c_1,c_2$ then in order that they match $T$, we must have $v_1 = T(e_1)+T(e_2)$ and $v_2=T(e_1)-T(e_2)$ otherwise we will be specifying a different transformation.

In general you can't just change the basis and keep the same 'w's.

Answer 2

Your comment is correct. the $T$ in question here was defined in terms of the basis given. So each basis would indeed have a different $T$

Answer 3

If $Tv=w$, that will be true in any basis. Given a basis, you can write out $T$, $v$, and $w$ in that basis, and $Tv=w$ will be true. Conversely, if $Tv=w$ is true in one basis, it will be true in all choices of basis.

A subspace can be defined as a span of a set of vectors. A basis for a subspace is a minimal spanning set. You need a basis to write explicit components for matrices and vectors.

You can define $T$ using components in a particular basis. The action of $T$ is coordinate independent; only the descriptions change, as you change basis.

You can nail down which transformation you are talking about, specify $T$ uniquely, by giving its matrix in some basis along with the choice of basis. If you give a random grid of numbers, there is no way to know what transformation that represents without assuming a basis, such as the standard basis.