# Review: Linear algebra
```{glossary}
inverse
Multiplicative inverse of a matrix.
invertible
Describing a matrix (necessarily square) that has an inverse.
```
An $m\times n$ matrix $\mathbf{A}$ is a rectangular $m$-by-$n$ array of numbers called **elements** or **entries**. The numbers $m$ and $n$ are called the **row dimension** and the **column dimension**, respectively; collectively they describe the **size** of $\mathbf{A}$. We say $\mathbf{A}$ belongs to $\mathbb{R}^{m\times n}$ if its entries are real or $\mathbb{C}^{m\times n}$ if they are complex-valued. A **square** matrix has equal row and column dimensions. A **row vector** has dimension $1\times n$, while a **column vector** has dimension $m \times 1$. By default, a vector is understood to be a column vector, and we use $\mathbb{R}^n$ or $\mathbb{C}^n$ to denote spaces of vectors. An ordinary number in $\mathbb{R}$ or $\mathbb{C}$ may be called a **scalar**.
We use capital letters in bold to refer to matrices, and lowercase bold letters for vectors. In this book, all vectors are column vectors---in other words, matrices with multiple rows and one column. The bold symbol $\boldsymbol{0}$ may refer to a vector of all zeros or to a zero matrix, depending on context; we use $0$ as the scalar zero only.
To refer to a specific element of a matrix, we use the uppercase name of the matrix *without* boldface, as in $A_{24}$ to mean the $(2,4)$ element of $\mathbf{A}$. To refer to an element of a vector, we use just one subscript, as in $x_3$. If you see a boldface character with one or more subscripts, then you know that it is a matrix or vector that belongs to a numbered collection.
We will have frequent need to refer to the individual columns of a matrix as vectors. Our convention, is to use a lowercase bold version of the matrix name, with a subscript to represent the column number. Thus, $\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_n$ are the columns of the $m\times n$ matrix $\mathbf{A}$. Conversely, whenever we define a sequence of vectors $\mathbf{v}_1,\ldots,\mathbf{v}_p$, we can implicitly consider them to be columns of a matrix $\mathbf{V}$. Sometimes we might write $\mathbf{V}=\bigl[ \mathbf{v}_j \bigr]$ to emphasize the connection.
The **diagonal** (main diagonal) of an $n\times n$ matrix $\mathbf{A}$ refers to the entries $A_{ii}$, $i=1,\ldots,n$. The entries $A_{ij}$ where $j-i=k$ are on a **superdiagonal** if $k>0$ and a **subdiagonal** if $k<0$. The diagonals are numbered as suggested here:
```{math}
\begin{bmatrix}
0 & 1 & 2 & \cdots & n-1 \\
-1 & 0 & 1 & \cdots & n-2 \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
-n+2 & \cdots & -1 & 0 & 1\\
-n+1 & \cdots & -2 & -1 & 0
\end{bmatrix}.
```
A **diagonal** matrix is one whose entries are all zero off the main diagonal. An **upper triangular** matrix $\mathbf{U}$ has entries $U_{ij}$ with $U_{ij}=0$ if $i>j$, and a **lower triangular** matrix $\mathbf{L}$ has $L_{ij}=0$ if $i