### Videos

- Leah Howard has two sets of videos for Math 251:
- 3Blue1Brown has a nice video series called
Essence of Linear Algebra.
In particular, I recommend:
- Chapter 1: Vectors, what even are they?
- Section 3.5: Linear combinations, span, and basis vectors

### My Current Lecture Notes

Here are the notes for Fall 2023:

- Section 1.1
- We are omitting Binary Vectors and Modular Arithmetic from Section 1.1.
- Section 1.2
- Cross Product
- Section 1.3
- Section 2.1
- A quick tutorial on Cramer's Rule for your ECET course
- Section 2.2
- Section 2.3
- Section 2.4
- Section 3.1
- Section 3.2
- Section 3.3
- Section 3.4
- Section 3.5
- Section 3.6
- Complex Numbers
- Section 4.1
- Section 4.2
- Section 4.3
- Section 4.4
- Section 5.1
- Section 5.2
- Section 5.3
- Section 5.4
- Section 7.3

Lecture notes from Fall, 2022 can be found here.

### Exam Review

### Handouts

- Section 1.1: Properties of Vectors
- Section 1.3: Lines and Planes
- Section 2.2: Homogeneous Systems
- Section 2.3: Gilles' Handout on the Row/Column Interpretation of a System of Equations
- Section 2.4: Applications
- Chapter 3: Matrix Properties
- Section 3.5: Change of Basis

### Further Web Resources

Matlab resources:

- GNU Octave is an open-source alternative to MATLAB
- Octave Online is a web-based version of Octave
- Gilles' handout on Linear Algebra with MATLAB

The Geogebra simulations shown in class can be found at:

- Section 1.2: Vector projection in 2D
- Cross product
- Section 1.3: Normal to plane
- Section 1.3: Vector equation of a plane
- Section 1.3: Vector equation of a line in 3D
- Section 2.2: Solutions for 3x3 systems
- Section 4.1: Eigenvectors and eigenvalues
- Section 5.3: Gram Schmidt process
- Section 7.3: Least-Squares Approximation

Some students have found the following Wolfram Demonstration useful: Gauss-Jordan Method

For RREF calculations, I particularly recommend Adrian Stoll's RREF Calculator. This one's particularly great because your matrix can include complex numbers.

Using Wolfram Alpha to check your calculations:

- Find the dot product of two vectors
- Find the cross product of two vectors
- Project vector A [1,2,3] onto vector B [0,1,0]
- RREF a matrix
- Multiply two matrices
- Find the inverse of a matrix

Try using the Linear Algebra Toolkit to check your calculations. I particularly recommend:

- Row operation calculator
- Calculating the inverse using row operations
- Finding a basis for the null space of a matrix

Some conceptual videos:

- Section 1.2: Zach Star's video on applications of the dot product: watch the first 2:30 min
- Section 2.2: LeiosOS's video on Gaussian Elimination has a neat visualization of what Gaussian elimination is actually doing
- Section 2.2: watch the first 2:23 minutes of Zach Star's video on matrix manipulation for two different ways to think about what it means to solve a system of equations
- Section 3.1:
Zach Star's video on applications of matrices:
from 1:15 to 4:36 to see why we can use the inverse of matrix A to solve A
**x**=**b** - Section 3.5: Zach Star's video on matrix manipulation explains the null space, row space, and column starting at 2:25 and ending at 9:00
- Section 4.1: Zach Star's video on eigenvectors and eigenvalues: watch the first 2:56 min for a nice conceptualization, including complex eigenvectors
- Section 4.3: 3Blue1Brown's video on eigenvectors and eigenvalues