A linear map can join any two vector spaces, not only lists of numbers: differentiating a polynomial, evaluating it at a point, transposing a matrix. Each respects addition and scaling, and each is decided completely by what it does to a basis.

A translator who renders every sentence faithfully also renders every paragraph faithfully, since a paragraph is built from sentences. A linear map sends combinations to combinations in the same way, so knowing it on a basis is knowing it everywhere.

Let $V$ and $W$ be vector spaces. A map $T:V\to W$ is linear when $T(\mathbf{u}+\mathbf{v})=T\mathbf{u}+T\mathbf{v}$ and $T(c\mathbf{u})=cT\mathbf{u}$ for all vectors $\mathbf{u},\mathbf{v}$ of $V$ and all scalars $c$. $V$ is its domain and $W$ its codomain. When $V=\mathbb{R}^n$ and $W=\mathbb{R}^m$ these are the maps of the chapter on matrices, each the map of an $m\times n$ matrix; the definition itself never mentions coordinates.

• Differentiation $D:\mathbb{P}_3\to\mathbb{P}_2$, $D(p)=p'$, is linear: and .

Take a basis $\mathbf{v}_1,\ldots,\mathbf{v}_n$ of $V$ and any vectors $$ of . Every is in exactly one way, and a linear map with must send it to . Conversely, that formula defines a map, because the weights are unique, and the map is linear, because coordinates respect the operations. So the images of a basis may be chosen freely, and then nothing else is free.

Every vector of $V$ is a combination of the basis, in exactly one way.

A linear map with $T\mathbf{v}_j=\mathbf{w}_j$ passes through the combination, so it has no choice.

Conversely, define $T$ by the formula: the weights are unique, so each $\mathbf{v}$ gets exactly one image.

Coordinates respect addition and scaling, so the map just defined is linear.

The images of a basis can be chosen freely, and then they decide the whole map.

The test is the same as in $\mathbb{R}^n$: does the map respect sums and multiples? Differentiation does, because the derivative of a sum is the sum of the derivatives.

If $T$ is linear and $\mathbf{v}_1,\ldots,\mathbf{v}_n$ is a basis, then $$. Knowing the images of a basis is knowing the whole map.

• $T:V\to W$ is linear when it respects addition and scaling; $V$ is the domain and $W$ the codomain.
• Differentiation, evaluation, transposition and multiplication by a matrix are linear.
• $T\mathbf{0}=\mathbf{0}$, and compositions of linear maps are linear.

Great work! You now know how to:

• test maps between polynomials, matrices and other spaces for linearity;
• name the domain and the codomain of a map;
• compute a linear map from its values on a basis;
• reject a map that moves the zero vector.