Why this proof about existence of a unique linear transformation is split into three parts?

Question

I was reading a proof, which the author split it into three parts. I didn't quite understand it when I first read it. And now I'm wondering how the author came up with the proof.

It seems like it's using the fact of (b) when defining the function $\textsf{T}$.

Can it be proved like this?:

Let $x\in\textsf{V}$, and $\textsf{T}(v_i)=w_i$ for $i=1,2,\dots,n$. Define $\textsf{T}:\textsf{V}\to\textsf{W}$ by $$\textsf{T}(x)=\sum^{n}_{i=1}a_i\textsf{T}(v_i).$$

Then I just have to prove that if I have another linear function $\textsf{U}$ such that $\textsf{U}(v_i)=w_i$ for $i=1,2,\dots,n$ then they're identical.

So can I prove it like this? And how can I practice this type of proof?

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Pictures part:

Answer 1

The author proves three properties (linearity, correct values at the $v_i$, uniqueness) because each of these properties is part of the claim.

"Then I just have to prove that if I have another linear function $U$ such that $U(v_i)=w_i$ ..."

Before you can say "another linear function" you better prove that your $T$ is linear. Before you say "another ... such that $U(v_i)=w_i$", you better prove that $T(v_i)=w_i$. Both these steops are done in parts a) and b) of the quoted proof. After that, indeed, all you have to show is uniqueness (as in part c).

Answer 2

The idea here is that the author is trying to prove that a unique linear application such that, with $(v_i)_{1\leq i \leq n}$ a basis of your v-space $V$, $\forall i \in [1..n], T(v_i) = w_i$ exists. The reason why there are three parts to his proof is that there are three independent parts to prove.

1. The first point is to prove that it is linear.

2. The second point is to prove that $\forall i \in [1..n], T(v_i) = w_i$.

3. The third point is to prove that it is unique, that's to say: If I can find two applications $T$ and $U$ that both satisfy conditions 1 and 2, then we have $T = U$.

This is a really common way to prove things, in particular in linear algebra, where applications usually have multiple properties to verify.

A textbook example is a dot product/scalar product, which is a bilinear, definite, positive form. This is 4 different properties to verify, and since these four parts are independent, you can (and usually will) prove each part separately.

So, in a sense, yes, you can prove things like that. But the proof of unicity should always come last.

So far as "how can I practice this type of proof?" is concerned, the answer is "Do linear algebra exercises". There are a plethora of exercises like that, where you're asked to prove that a unique linear application such that [insert condition] exists. You can also prove that a given application belongs to a certain type of applications, like scalar products or so on. Linear algebra is full to the brim with these kinds of exercises.