Requirements for a Linear Transformation

Let's say we have a transformation:

$$T: \mathbb{R}^n \rightarrow \mathbb{R}^m.$$

This is a linear transformation iff: for all $\vec{a} , \vec{b} \in \mathbb{R}^n$ and $c \in \Bbb{R}$,

1. $T(\vec{a} + \vec{b}) = T(\vec{a}) + T(\vec{b})$
2. $T(c \vec{a}) = cT(\vec{a})$

I've seen this kind of 'requirements' multiple times in Linear Algebra, and I wonder what the names for these requirements are.

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2 Answers

1. $T$ preserves addition
2. $T$ preserves multiplication by scalar

Altogether: $T$ is linear.

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Another way to say this is that $T$ is additive and homogeneous of degree $1$. In general, a function $f:X \to Y$ is additive if $f(x+y)=f(x)+f(y)$ for all $x,y \in X$. If $X$ and $Y$ are vector spaces then we say that $f$ is homogeneous of degree $k$ if for all $a$ not equal to $0$ in the underlying scalar field of $X$ and all $x \in X$, $f(ax)=a^kf(x)$ for some integer $k$.

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I really think that the richness in your answer elegantly equals the conciseness and thorough nature of Berci's answer. Nice job! – 000 Oct 21 '12 at 13:58
Thanks, I appreciate that! – Alex Petzke Oct 21 '12 at 17:31