We study functions $T:V \to W$ which have the special property that $T(x + y) = T(x) + T(y)$ and $T(cx) = c T(x)$ for all $x, y \in V$ and scalars $c$. The spaces $V$ and $W$ are assumed to have just enough structure that these equations make sense (in other words $V$ and $W$ are what we call vector spaces). Often $V = \mathbb R^n$ and $W= \mathbb R^m$.

We would like to understand these types of functions as well as possible. An important strategy is to find a special list of vectors $v_1, \ldots, v_n \in V$ such that understanding what $T$ does to $v_i$ is easy. For example, maybe $T(v_i) = \lambda_i v_i$ for some scalar $\lambda_i$. If we are lucky, then any vector $v \in V$ can be written as a combination of the special vectors $v_i$, and if so then that makes it easier to evaluate $T(v)$. This viewpoint helps us to understand $T$ and do calculations with $T$ easily.

In linear algebra we study “linear transformations” which are functions $T:V \to W$ which have the special property that $T(x + y) = T(x) + T(y)$ and $T(cx) = c T(x)$ for all $x, y \in V$ and scalars $c$. The spaces $V$ and $W$ are assumed to have just enough structure that these equations make sense (in other words $V$ and $W$ are what we call vector spaces). Often $V = \mathbb R^n$ and $W= \mathbb R^m$.

We would like to understand these types of functions as well as possible. An important strategy is to find a special list of vectors $v_1, \ldots, v_n \in V$ such that understanding what $T$ does to $v_i$ is easy. For example, maybe $T(v_i) = \lambda_i v_i$ for some scalar $\lambda_i$. If we are lucky, then any vector $v \in V$ can be written as a combination of the special vectors $v_i$, and if so then that makes it easier to evaluate $T(v)$. This viewpoint helps us to understand $T$ and do calculations with $T$ easily.

One reason linear algebra is important is that the fundamental strategy of calculus is to take a nonlinear function (difficult) and approximate it locally by a linear function (easy).