Matrices, Linear Maps, and Determinants

Definition An $m \times n$ matrix $A$ is a rectangular array of real numbers with $m$ rows and $n$ columns. The entry in row $i$ and column $j$ is written $a_{ij}$. A vector in $\mathbb{R}^n$ is written as a column, an $n \times 1$ matrix. If the columns of $A$ are the vectors $\mathbf{a}_1, \dots, \mathbf{a}_n$ in $\mathbb{R}^m$, we write

Two matrices are equal when they have the same size and equal entries in every position. The $n \times n$ identity matrix $I_n$ has $1$ on the diagonal and $0$ elsewhere. The transpose $A^{T}$ of an $m \times n$ matrix is the $n \times m$ matrix with entries $(A^{T})_{ij} = a_{ji}$, so the rows of $A$ become the columns of $A^{T}$.

Definition Let $A$ be an $m \times n$ matrix with columns $\mathbf{a}_1, \dots, \mathbf{a}_n$ and let $\mathbf{x} = \langle x_1, \dots, x_n \rangle$. The product $A\mathbf{x}$ is the vector in $\mathbb{R}^m$

Equivalently, its $i$th entry is $a_{i1} x_1 + \cdots + a_{in} x_n$, the dot product of row $i$ with $\mathbf{x}$. The product is defined only when $\mathbf{x}$ has as many entries as $A$ has columns. A map $T$ from $\mathbb{R}^n$ to $\mathbb{R}^m$ is linear if $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ and $T(c\mathbf{u}) = cT(\mathbf{u})$ for all vectors $\mathbf{u}, \mathbf{v}$ and scalars $c$. Every map $\mathbf{x} \mapsto A\mathbf{x}$ is linear, and every linear map $T$ equals $\mathbf{x} \mapsto A\mathbf{x}$ for exactly one matrix, its standard matrix, whose $j$th column is $T(\mathbf{e}_j)$.

Definition Let $A$ be $m \times n$ and $B$ be $n \times p$, with columns $\mathbf{b}_1, \dots, \mathbf{b}_p$. The product $AB$ is the $m \times p$ matrix with columns $A\mathbf{b}_1, \dots, A\mathbf{b}_p$, so its entries are

It is defined only when the number of columns of $A$ equals the number of rows of $B$. With this definition $(AB)\mathbf{x} = A(B\mathbf{x})$ for every $\mathbf{x}$ in $\mathbb{R}^p$, so $AB$ is the matrix of the composition that applies $B$ first and then $A$. The product is associative, $(AB)C = A(BC)$, but in general $AB \ne BA$.

Definition A system of $m$ linear equations in $n$ unknowns,

is the single matrix equation $A\mathbf{x} = \mathbf{b}$, where $A = [a_{ij}]$ is the coefficient matrix. The system has a solution exactly when $\mathbf{b}$ is a linear combination of the columns of $A$. Its solution set is empty, a single point, or infinite. If $\mathbf{x}_p$ is one solution, the solutions are exactly the vectors $\mathbf{x}_p + \mathbf{h}$ with $A\mathbf{h} = \mathbf{0}$.

Definition A square $n \times n$ matrix $A$ is invertible if there is an $n \times n$ matrix $B$ with $AB = BA = I_n$. Such a $B$ is unique and is written $A^{-1}$. For a square matrix the following are equivalent. The matrix $A$ is invertible. The equation $A\mathbf{x} = \mathbf{b}$ has exactly one solution for every $\mathbf{b}$, namely $\mathbf{x} = A^{-1}\mathbf{b}$. Only $\mathbf{x} = \mathbf{0}$ solves $A\mathbf{x} = \mathbf{0}$. The columns are linearly independent. The determinant is not zero. For a $2 \times 2$ matrix with $ad - bc \ne 0$,

If $A$ and $B$ are invertible of the same size, then $(AB)^{-1} = B^{-1}A^{-1}$.

Definition The determinant of a $2 \times 2$ matrix is $ad - bc$, as above. For a $3 \times 3$ matrix $A = [a_{ij}]$, expansion along the first row gives

For a larger square matrix the determinant is defined the same way, by expanding along the first row into smaller determinants with alternating signs. For square matrices $A$ and $B$ of the same size, $\det(AB) = \det A \det B$ and $\det A^{T} = \det A$. Swapping two rows changes the sign, multiplying one row by a number $k$ multiplies the determinant by $k$, adding a multiple of one row to another changes nothing, and $A$ is invertible exactly when $\det A \ne 0$. Because $\det A^{T} = \det A$, the same rules hold for columns. For vectors $\mathbf{u}, \mathbf{v}, \mathbf{w}$ in $\mathbb{R}^3$, the matrix with rows $\mathbf{u}, \mathbf{v}, \mathbf{w}$ has determinant $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$, the scalar triple product.

Definition Let $A$ be a $2 \times 2$ matrix. For every bounded region $R$ in the plane that has an area, for example one bounded by finitely many smooth curves, the image $A(R)$ has area

For a $3 \times 3$ matrix the same holds with volume in place of area. In particular the unit square goes to the parallelogram spanned by the columns, with area $\lvert \det A \rvert$, and the unit cube goes to the parallelepiped spanned by the columns, with volume $\lvert \det A \rvert$. When $\det A = 0$ the image is flattened and has area or volume $0$.

Definition An ordered basis $(\mathbf{v}_1, \mathbf{v}_2)$ of $\mathbb{R}^2$ is positively oriented if the matrix with columns $\mathbf{v}_1, \mathbf{v}_2$ has positive determinant, and negatively oriented if the determinant is negative. An ordered basis $(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3)$ of $\mathbb{R}^3$ is positively oriented, or right-handed, when

The standard bases are positively oriented, since $\det I = 1$. An invertible linear map preserves orientation when $\det A > 0$ and reverses it when $\det A < 0$.

Definition A basis of $\mathbb{R}^n$ is a list of $n$ vectors $\mathbf{b}_1, \dots, \mathbf{b}_n$ for which the matrix $P = [\mathbf{b}_1 \ \cdots \ \mathbf{b}_n]$ is invertible. Every $\mathbf{x}$ in $\mathbb{R}^n$ then has unique coordinates $\mathbf{c} = [\mathbf{x}]_{\mathcal{B}}$ with $\mathbf{x} = c_1\mathbf{b}_1 + \cdots + c_n\mathbf{b}_n$, that is

If a linear map has standard matrix $A$, then in $\mathcal{B}$ coordinates it has matrix $P^{-1}AP$, meaning $[A\mathbf{x}]_{\mathcal{B}} = P^{-1}AP\,[\mathbf{x}]_{\mathcal{B}}$. To convert from $\mathcal{B}$ coordinates to coordinates in a second basis $\mathcal{C}$ with matrix $Q$, first return to standard coordinates and then solve, so $[\mathbf{x}]_{\mathcal{C}} = Q^{-1}P\,[\mathbf{x}]_{\mathcal{B}}$.