Array vs. matrix vs. 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.
Definitions
An array is an arrangement of objects in rows and columns.
An array is a set of items arranged in rows and columns (or in a list, or in a block with more directions).
An array is a collection of elements indexed by tuples of integers $(i_1, \dots, i_k)$ in a rectangular index range. The number $k$ of indices is the number of axes (or dimensions) of the array.
Examples
-
The matrix \(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\)
It represents rotation by $90°$ in the standard basis, but it is a grid of numbers, not a function.
Non-examples
-
Rotation of the plane by $90°$
It is a function. Different bases give different matrices for it.
Think of numbers arranged in a grid, like the seats in a theater. A matrix is a rectangular grid of numbers arranged in rows and columns.
A matrix is a rectangular grid of numbers arranged in rows and columns. A matrix can be multiplied by a scalar, two matrices $A$ and $B$ of the same size can be added ($A + B$), and two matrices $A$ and $B$ can be multiplied ($AB$) if $A$ and $B$ have compatible sizes.
An $m \times n$ matrix over a field $F$ (such as $\reals$) is an array $A = (a_{ij})$ with entries in $F$, together with the operations of scalar multiplication, matrix addition, and matrix multiplication.
Examples
-
The matrix \(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\)
It represents rotation by $90°$ in the standard basis, but it is a grid of numbers, not a function.
Non-examples
-
Rotation of the plane by $90°$
It is a function. Different bases give different matrices for it.
-
Differentiation, on polynomials of degree at most $2$
In the basis $(1, x, x^2)$, its matrix has rows $(0, 1, 0)$, $(0, 0, 2)$, and $(0, 0, 0)$.
Consider a function $f$ from $\reals$ to $\reals$. The function $f$ is a linear map if it 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.
Consider vector spaces $V$ and $W$, and a function $T : V \to W$. The function $T$ is called a linear map if $T(u + v) = T(u) + T(v)$ and $T(c v) = c T(v)$ for all vectors $u, v$ and scalars $c$.
Examples
-
Rotation of the plane by $90°$
It is a function. Different bases give different matrices for it.
-
Differentiation, on polynomials of degree at most $2$
In the basis $(1, x, x^2)$, its matrix has rows $(0, 1, 0)$, $(0, 0, 2)$, and $(0, 0, 0)$.
Non-examples
-
The matrix \(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\)
It represents rotation by $90°$ in the standard basis, but it is a grid of numbers, not a function.
Examples
| Example | array | matrix | linear map | Why |
|---|---|---|---|---|
Missing example: a_3_by_3_grid_of_names | ||||
Missing example: a_2_by_3_by_4_block_of_numbers | ||||
| The matrix \(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\) | ✓ | ✓ | ✗ | It represents rotation by $90°$ in the standard basis, but it is a grid of numbers, not a function. |
| Rotation of the plane by $90°$ | ✗ | ✗ | ✓ | It is a function. Different bases give different matrices for it. |
| Differentiation, on polynomials of degree at most $2$ | – | ✗ | ✓ | In the basis $(1, x, x^2)$, its matrix has rows $(0, 1, 0)$, $(0, 0, 2)$, and $(0, 0, 0)$. |
Notes
- A matrix $A$ defines the linear map $x \mapsto A x$ from $\reals^n$ to $\reals^m$, and every linear map between these spaces has this form. This is why matrices and linear maps are easily identified.
- For a linear map $T : V \to V$, changing the basis by an invertible matrix $P$ changes its matrix from $A$ to $P^{-1} A P$. Properties of $T$ (such as eigenvalues, trace, and determinant) are those that all of these matrices share.