Is it ok to combine vector addition and scalar multiplication when proving linear transformations?

Question

So we're on the topic of eigenvectors and this came up the other day.

If $T(c$₁$v$₁+$c$₂$v$₂$)=c$₁$T(v$₁$)+c$₂$T(v$₂$)$ for all vectors $v$₁ and $v$₂ in $V$ and scalars $c$₁ and $c$₂, then $T$ is a linear transformation.

Proving is not really something I have experience with so I'm not sure if this is ok.

Answer 1

We say a map ϕ : X → Y between two vector spaces X, Y is called a linear map if the following holds:

ϕ(λ·$x_1$+µ·$x_2$) =λ·ϕ($x_1$)+µ·ϕ($x_2$) for all vectors $x_1$, $x_2$ in X and real numbers λ, µ.

Answer 2

Sometimes the condition you've stated is the definition of a linear transform. Since you're confused about that, maybe you've seen an alternate definition which states the following:

• $T(av) = aT(v)$ for every scalar $a$ and vector $v \in V$
• $T(v + w) = T(v) + T(w)$ for every pair of vectors $v, w \in V$

It can be seen the condition given in your question implies the above. Set $c_1 = a$ and $c_2 = 0$ first and then, $c_1 = c_2 = 1.$
In fact, the two conditions are equivalent, it should be easy to see how the above definition implies the conditions given in your question.