Linear map
High school
Consider a function $f$ from $\reals$ to $\reals$. The function $f$ is a linear map if $f$ has the form $f(x) = m x$ for some number $m$. The graph of a linear map is a straight line that passes through the origin.
Caveat. In high school algebra, $f(x) = m x + b$ is called a “linear function,” but it is a linear map only when $b = 0$. With $b \neq 0$, it is called affine.
Undergraduate
Consider vector spaces $V$ and $W$, and a function $T : V \to W$. The function $T$ is called a linear map if, for all vectors $u, v$ and scalars $c$,
- $T(u + v) = T(u) + T(v)$ (additivity), and
- $T(c v) = c T(v)$ (homogeneity).
Examples
Examples
-
Rotation of the plane by $90°$
High school
Rotation of the plane by $90°$ is a function, and different bases give different matrices for it.
- A linear map $T : \reals^2 \to \reals^2$ Undergraduate
-
Differentiation, on polynomials of degree at most $2$
Undergraduate
In the basis $(1, x, x^2)$, the matrix of differentiation has rows $(0, 1, 0)$, $(0, 0, 2)$, and $(0, 0, 0)$.
- An invertible linear map $V \to V$ Undergraduate
- A linear map $\reals^2 \to \reals^3$ Undergraduate
Non-examples
Things that may be mistaken for examples.
-
The matrix \(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\)
High school
The matrix represents rotation by $90°$ in the standard basis, but it is a grid of numbers, not a function.
Distinctions involving this term
- Array vs. matrix vs. linear map: Grids of numbers appear throughout linear algebra and computing, but a grid can be just data, a matrix with its own arithmetic, or a way of writing a linear map. An array is a grid of data. A matrix is a grid of numbers with matrix arithmetic. A linear map is a function between vector spaces. A matrix represents a linear map as an array of scalars with respect to a given vector basis.