Vectors

A vector is just a list of numbers. Given a list $b_1,\ldots,b_n$ we usually write a vector vertically $\begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{pmatrix}$ as in a single column. We often write $\mathbf{v}$ to denote a vector. The entries of $\mathbf{v}$ are called the components of $\mathbf{v}$.

The length of the $\mathbf{v}$ is number of entries in $\mathbf{v}$.

A scalar is another name for a vector of length 1. We typically omit the parentheses when writing a scalar.

We have some basic, but useful, operations we can do with vectors.

Vector addition

If $\mathbf{v}$ and $\mathbf{w}$ are vectors of the same length, we can add them. We do this by adding their components. More precisely, if $\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_m \end{pmatrix} \ , \ \mathbf{w} = \begin{pmatrix} w_1 \\ w_2 \\ \vdots \\ w_m \end{pmatrix}$ then $\mathbf{v} + \mathbf{w} := \begin{pmatrix} v_1 + w_1 \\ v_2 + w_2 \\ \vdots \\ v_m + w_m \end{pmatrix}$

For example, $\begin{pmatrix} -1 \\ 4 \\ 3 \end{pmatrix} + \begin{pmatrix} 5 \\ -6 \\ 0 \end{pmatrix} = \begin{pmatrix} 4 \\ -2 \\ 3 \end{pmatrix}$

Note: We cannot add vectors of different lengths.

Vectors and multiplication

There are standard ways to multiply vectors but, like with addition, there are some restrictions on the operations.

Scalar multiplication

Given a scalar $a$ and a vector $\mathbf{v}$ we get another vector $a \mathbf{v}$ as $a \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_m \end{pmatrix} = \begin{pmatrix} av_1 \\ av_2 \\ \vdots \\ av_m \end{pmatrix}$ In other words, we scale each component of $\mathbf{v}$ by $a$. For example, $3 \begin{pmatrix} -1 \\ 2 \\ 0 \end{pmatrix} = \begin{pmatrix} -3 \\ -6 \\ 0 \end{pmatrix}$

Multiplication of vectors

Given two vectors $\mathbf{v}$ and $\mathbf{w}$ of the same length, we can multiply them and return a scalar. When we write the formula, we do something a little strange. We turn the first vector into a row, in the notation.

The formula is $\begin{pmatrix} v_1 & v_2 & \cdots & v_m \end{pmatrix} \begin{pmatrix} w_1 \\ w_2 \\ \vdots \\ w_m \end{pmatrix} := \sum_{i=1}^m v_i w_i.$

For example,

$ \begin{pmatrix} 1 & 2 & 3 & 4 \end{pmatrix} \begin{pmatrix} -1 \\ 0 \\ 1 \\ 2 \end{pmatrix} = 1(-1) + 2(0) + 3(1) + 4(2) = 10. $

Next, we look at matrices.