### Introduction to matrices

You can view a matrix simply as a generalization of a vector, where we arrange numbers in both rows and columns. Let's keep the number of rows and columns arbitrary, letting $m$ be the number of rows and $n$ the number of columns. We refer to such a matrix as an $m \times n$ matrix and write it as \begin{align*} A= \left[ \begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1n}\\ a_{21} & a_{22} & \ldots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & \ldots & a_{mn} \end{array} \right]. \end{align*} For example, a $3 \times 2$ matrix is \begin{align*} B= \left[ \begin{array}{rr} 4 & -3\\ 7 & 9\\ -5& 0 \end{array} \right], \end{align*} and a $4 \times 7$ matrix is \begin{align*} C= \left[ \begin{array}{rrrrrrr} 9 & -9 & -8 & 1 & 4 & -3 & -3\\ 7 & -1 & 7 & -3 & -5& -2& 9\\ 11 & 1 & 8 & -5& -5 & 0 & -2\\ 9 & -2 & -8 & -1 & 3 & 10 & 0 \end{array} \right]. \end{align*}

The arrangement of a matrix in rows and columns is more than just to make it look pretty. The structure of a matrix allows us to define a fundamental operation on matrices: multiplication. This multiplication forms the basis of linear algebra. In particular, this matrix multiplication allows matrices to represent linear transformations (or linear functions) that transform vectors into other vectors. (A simple example of a linear transformation is the rotation of a vector.) Other uses of matrices involve calculating their determinant.

#### Vectors as matrices

The concept of matrices is so powerful, that in many cases, we make our lives simpler by viewing a vector as a special type of matrix. By comparing a vector such as $\vc{x}=(1,5,3)$ to a matrix, it initially seems that the difference between vectors and matrices is that vectors have only one row while matrices have multiple rows. However, there is one important twist (literally) that isn't apparent when writing vectors in the form $\vc{x}=(1,5,3)$ . When we view vectors as matrices, we actually view them as a rotated version of the standard form, writing an $n$-dimensional vector as a $n \times 1$ column matrix \begin{align*} \vc{x} = \left[ \begin{array}{c} x_1\\ x_2\\ x_3\\ \vdots\\ x_n \end{array} \right]. \end{align*} We often call $\vc{x}$ an $n\times 1$ column vector and use the terms “column vector” and “column matrix” synonymously. The vector $\vc{x}=(1,5,3)$ written as a $3 \times 1$ column vector would be \begin{align*} \vc{x} = \left[ \begin{array}{c} 1 \\ 5 \\ 3 \end{array} \right]. \end{align*}

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