MANE 3351
Lecture 20
Classroom Management
Agenda
- Summary of operations on Vectors, Matrix Multiplication, Three-Dimensional Transformations
- Determinants
- Introduction to Inverse Matrix
- Lab Assignment 9 today
Resources
Handouts
Calendar
| Date | Lecture Topic | Lab Topic |
|---|---|---|
| 11/11 | Lecture 20 - Determinant | Lab 9 |
| 11/13 | ||
| 11/18 | ||
| 11/20 | ||
| 11/25 | ||
| 11/27 | ||
| 12/2 | ||
| 12/4 | ||
| 12/9 | Final exam 1:15 - 3:00 pm |
Lecture 20 Overview
- Vector Operations,
- Determinants, and
- Matrix Inversion
Vectors
A vector \(\mathbf{x}\in{\Bbb R}^n\)
\[
\mathbf{x}=\begin{bmatrix}x_1\\x_2\\ \vdots \\ x_n
\end{bmatrix}
\]
can be thought of as a one-dimensional array of numbers and is written as
- \(\mathbf{x}\) is often called a column vector
- the dimension of \(\mathbf{x}\) is \((n\times 1)\)
A row vector can be written as
\[
\mathbf{y}=\begin{bmatrix}y_1\;y_2\;\cdots\;y_m\end{bmatrix}
\]
- the dimension of \(\mathbf{y}\) is \((1\times m)\)
Scalar Product
\[
\alpha\mathbf{x}=\begin{bmatrix}\alpha x_1\\ \alpha_2 x_2 \\ \vdots \\ \alpha x_n\end{bmatrix}
\]
for \(\alpha\) (a constant or scalar)
Addition/Subtraction
\[
\mathbf{x}\pm\mathbf{y}=\begin{bmatrix}x_1\pm y_1\\ x_2\pm y_2\\ \vdots \\ x_n\pm y_n\end{bmatrix}
\]
- Note that the dimensions of \(\mathbf{x}\) and \(\mathbf{y}\) must be identical
Matrices
A matrix is a two-dimensional array of numbers written as
\[
\mathbf{A}=\begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1m}\\a_{21}&a_{22}&\cdots&a_{2m}\\ \vdots&\vdots&\ddots&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nm}\end{bmatrix}
\]
- A matrix has two dimensions and can be written as \(A_{n\times m}\) where \(n\) is the number of rows and \(m\) is the number of columns
- A column vector can be considered an \(n\times 1\) matrix and a row vector can be considered an \(1\times m\) matrix
- Matrices may or may not be square