MANE 3351
Lecture 20
Classroom Management
Agenda
- Summary of operations on Vectors, Matrix Multiplication, Three-Dimensional Transformations
- Determinants
- Introduction to Inverse Matrix
- Lab Assignment 8 (assigned 11/11/25, due 11/18/25 before 9:30 AM)
Resources
Handouts
Calendar
| Week | Monday Lecture | Wednesday Lecture |
|---|---|---|
| 11 | 11/10: Test 2 | 11/12: Lecture 20 |
| 12 | 11/17: Lecture 21 | 11/19: Lecture 22 |
| 13 | 11/24: Lecture 23 | 11/26: Lecture 24 |
| 14 | 12/1: Lecture 25 | 12/3: Lecture 26 |
| 15 | 12/8: Lecture 27 | 12/10: Review |
Final Exam is Monday 12/15/2025 8:00 - 9:45 AM
I will be off-campus participating in an ABET visit and a proctor will be arranged for the final exam.
Lecture 20 Overview
- Vector Operations,
- Determinants, and
- Matrix Inversion
Vectors
A vector \(\mathbf{x} \in \mathbb{R}^n\)
\[
\mathbf{x}=\begin{bmatrix}x_1\\x_2\\ \vdots \\ x_n
\end{bmatrix}
\]
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