Matrix
Elementary
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.
High school
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.
Undergraduate
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
Examples
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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.
Non-examples
Things that may be mistaken for examples.
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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.
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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 $(0, 0, 0)$.
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.