Vector and Matrix

May 26, 2019

These are some useful points that Prof. Gilbert did not mention in the class but still I think that will help you get started better, in case you did not know.

Vector

You are probably familiar with vectors from Calculus. That’s the column vector. And in my notes I use brackets to represent vectors. In some notes you may see they use parenthesis.

$u=\begin{bmatrix} 2\\ 1 \end{bmatrix},v=\begin{bmatrix} \pi\\ e \end{bmatrix}$

Bold characters are used to represent unknown variable vector, sometimes short for unknown vector, to differentiate with single variable unknown.

$\mathbf{x}=\begin{bmatrix} x_1\\ x_2\\ x_3\\ ... \end{bmatrix}$

Row then column

In convention, the subscript of a matrix’s item $a_{ij}$ is ith row jth column. When I say a matrix is $4\times3$ or in $\mathbb R^{4\times3}$ , it means it has 4 rows and 3 columns.

Matrix

You have probably known that marix muliplication in this form:

$c_{ij}=\sum_{k=1}^Na_{ik}b_{kj}$

An item in the product matrix is equal to a row in the left matrix times a column in the right matrix. What is useful to remember before the lecture notes or the video series is:

$\begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9\\ \end{bmatrix}\begin{bmatrix} x\\ y\\ z \end{bmatrix}=\begin{bmatrix} 1x+2y+3z\\ 4x+5y+6z\\ 7x+8y+9z \end{bmatrix}=x\begin{bmatrix} 1\\ 4\\ 7 \end{bmatrix}+y\begin{bmatrix} 2\\ 5\\ 8 \end{bmatrix}+z\begin{bmatrix} 3\\ 6\\ 9 \end{bmatrix}$

we can rewrite the multiplication as a combination of the left matrix’s columns. This is true for any matrix multiplication. The situations the vector $\begin{bmatrix} x\\ y\\ z \end{bmatrix}$ has more than one column will be illustrated in later course notes.