On proving that a map is linear

Question

Let $T:V\to W$ be a map on a vector space. To show that it is linear, one must show given $\vec{v},\vec{w}\in V$ and $a\in \mathbf{F}$ where $\mathbf{F}$ is the field which $V$ is defined over, that

• $T(\vec{v}+\vec{w})=T(\vec{v})+T(\vec{w}) \qquad $ (Additivity)
• $T(a\vec{v}) = a T(\vec{v}) \qquad $ (Homogeneity)

My question is whether the following equivalencies are sufficient prove linearity alone (that is, not to show additivity and homogeneity separately):

• $T(a\vec{v}+b\vec{w}) = aT(\vec{v})+bT(\vec{w})$

or

• $T(a\vec{v}+\vec{w}) = aT(\vec{v})+T(\vec{w})$

or

• $T(a(\vec{v}+\vec{w})) = aT(\vec{v})+aT(\vec{w})$

I ask this because sometimes it can be tedious to show both properties separately when the map is complicated.

Furthermore, when does it suffice to combine two properties into one statement and proving that statement also proves the properties which it is composed of?

Thanks

Answer 1

Your second version, $T(a\vec{v}+\vec{w}) = aT(\vec{v})+T(\vec{w})$, is equivalent and is typically the most convenient form for proofs.

In order to see that a function satisfying the equation $T(a\vec{v}+\vec{w}) = aT(\vec{v})+T(\vec{w})$ (for all vectors $\vec v, \vec w$ and scalars $a$) is indeed linear, note that setting $\vec w = \vec 0$ shows that $T$ is homogeneous and setting $a = 1$ shows that $T$ is additive.