- The set of \(m\times n\) matrices with real entries is denoted \(\mathbb{R}^{m\times n}\).
- Given a matrix \(A\in\mathbb{R}^{m\times n}\), the column space \(C(A) = \{Ab : b\in\mathbb{R}^{n}\}\) is sometimes called the
**image**of \(A\) and denoted \(\operatorname{im}(A)\). - A vector in \(\mathbb{R}^{n}\) can either be written as a column vector: \[\left[\begin{array}{r} x_{1}\\ x_{2}\\ \vdots\\ x_{n}\end{array}\right]\] or as an \(n\)-tuple: \[(x_{1},x_{2},\ldots,x_{n}).\] For example, \[\left[\begin{array}{r} 2\\ -3\end{array}\right]\quad\text{and}\quad (2,-3)\] denote the same vector.

- Given a matrix \(A\in\mathbb{R}^{m\times n}\), the null space \(N(A) = \{x\in\mathbb{R}^{n} : Ax=0\}\) is sometimes called the
**kernel**of \(A\) and denoted \(\operatorname{ker}(A)\).