Linear algebra/Linear maps

A function $$f:U\rightarrow V $$ from a vector space $$U$$ to a vector space $$V$$ is called a linear map if

1) For all $$\mathbf{v_1},\mathbf{v_2}\in U$$, $$f(\mathbf{v_1}+\mathbf{v_2}) =  f(\mathbf{v_1}) + f(\mathbf{v_2})$$

2) For all $$c\in F$$ and all $$\mathbf{v} \in U$$, $$f(c\mathbf{v}) = cf(\mathbf{v})$$.

These two algebraic properties have similar analogs in other branches of algebra as well. The recurring theme is "compatibility", in the sense that the function $$A$$ is "compatible" with the operations of vector addition and scalar multiplication. The significance of these properties may not be clear to novice readers, but will gradually become clearer to the persistent scholar. For now, it is important to simply keep them in mind. Linear maps on real or complex vector spaces are among the most frequently studied. For example, linear functions from $$\mathbb{R}$$ to $$\mathbb{R}$$ with no constant term are all linear maps. These functions have the form $$f(x) := mx$$ for some $$m\in \mathbb{R}$$.

The first property follows easily from the distributivity of multiplication:

$$ \begin{align} f(p + q) &:= m(p + q) \\ &:= mp + mq \\ &:= f(p) + f(q) \end{align} $$

The second property is left to the reader.

Given a matrix $$M \in \mathbb{R}^{n\times m}$$, the function $$A:\mathbb{R}^{m} \rightarrow \mathbb{R}^{n} $$ defined by the matrix product $$A(v) := Mv $$ is also a linear map. The proof is left to the reader.