Definition 1.1.1. Matrix.
A matrix is a rectangular array of numbers. If this array has \(m\) rows and \(n\) columns, we say the matrix is an \(m\times n\) matrix.
Section 1.1 Matrices, Vectors and Linear Combinations
The primary objects of study in the field of linear algebra and its applications are linear transformations between vector spaces. These linear transformations are often represented using matrices.
The following are examples of matrices:
\begin{equation*}
\begin{bmatrix} 1 & 2 \\ -3 & 4 \end{bmatrix}\text{ and }\begin{bmatrix}1 & -2 & 0 \\ 3 & -21 & 2\end{bmatrix}
\end{equation*}
The first is \(2\times2\) and the second is \(2\times3\text{.}\)
We say that a matrix is a square matrix if it has the same number of rows as columns.
\begin{equation*}
A = [a_{ij}] = \begin{bmatrix}
a_{11} & a_{12} & \dots & a_{1n} \\
a_{21} & a_{22} & \dots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \dots & a_{nn}
\end{bmatrix},
\end{equation*}
The diagonal entries are \(a_{ii},1\leq i\leq n\) and these form the main diagonal of the matrix.
As important as matrices are in applications of mathematics, many computing solutions exist for handling computations involving them. One open source solution (included in Sage/CoCalc!) is Octave 1 , which is a free alternative to MATLAB. In the code cell below Octave is used to define the square matrix above and get its diagonal entries. Note that brackets must be used to contain the entries of the matrix, entries in the same row must be separated by commas (or spaces) and rows are separated by semicolons.
A matrix with only one column or one row is known as a vector. We will often use vectors to represent points and matrices to represent transformations of points. As a first step to this, we can define addition and scalar multiplication for matrices.
Definition 1.1.2. Matrix Addition and Scalar Multiplication.
Let \(A\) and \(B\) denote matrices of the same size and let \(c\in\RR\text{.}\) Then \(A+B\) is the matrix obtained by adding the entries of \(A\) and \(B\) componentwise. Likewise, \(cA\) is the matrix obtained by multiplying the entries of \(A\) by \(c\) componentwise.
Theorem 1.1.3. Properties of Matrix Addition and Scalar Multiplication.
Let \(A, B\) and \(C\) denote matrices of the same size and let \(r,s\in\RR\text{.}\) Then
- \(A+B = B+A\text{.}\)
- \((A+B)+C = A+(B+C)\text{.}\)
- \(A+\mathbf{0} = A\) where \(\mathbf{0}\) denotes the zero matrix.
- \(r(A+B) = rA + rB\text{.}\)
- \((r+s)A = rA + sA\text{.}\)
- \(r(sA) = (rs)A\text{.}\)
- \(1A = A\text{.}\)
- \(0A = \mathbf{0}\text{.}\)
https://www.gnu.org/software/octave/index
