Extra Resources for APSC 174

Dont worry if youre feeling lost in APSC 174, all of the profs have difficulties effectively teaching the course material. These resources will be provided as a way of augmenting our learning in ways that hopefully make more sense than the lectures we are given.

This is not a replacement of attending lectures!! There is material that is covered in class that is so weird and niche that I won't be able to find resources for it.

Organised by section in the lecture notes.

If you want more on a particular topic, or have resources to share, post in the discussion tab or, if you're really desperate, send me an email at my email

Important Notes

We are taught if a transformation is "injective" or "surjective" (or bijective, but thats just both), these are sometimes refered to as "one-to-one" and "onto" respectively.

We are taught the "Image" and "Kernel" of a linear transformation. In some of these resources, these are refered to as the "Column Space" and "Null Space" respectively.

Ableson's Lectures

Ableson has posted recorded lectures! They're very good, I would recommend watching them if you're stuck.

• Week 2
• Week 3
• Week 4
• Week 5
• Week 6
• Week 7
• Week 8
• Week 10
• Week 11
• Week 12

For annotated notes (not mine, but very good), please email.

Previous Exams

Gradebank resources can be found here

General Resources

• Linear Algebra 4th Edition
  • The textbook for this course, useful if you skip lectures, but once again, probably not a complete replacement.
  • Solutions to problems can be found at the back of the textbook, but more slightly more detailed answers can be found HERE
• The Bright Side of Mathematics | Linear Algebra
  • Seems to cover a lot of the material that we do, and from watching some of his other content these vides should be very high quality.
• The Bright Side of Mathematics | Abstract Linear Algebra
  • Some of the topics in this playlist overlap with our content
• MIT OpenCourseWare: 18.06SC
  • This course is very hands on compared to what we do, a lot of the first lectures involving matrix decomposition is irrelevant to us.
  • There isn't a ton of overlap between these courses, but it is taught very well
• 3Blue1Brown: Essence of Linear Algebra
  • A great math youtube channel in general, and this series provides excellent visualizations of some of the weirder concepts.
• Khan Academy | Linear Algebra
  • If you don't know Khan Academy, you should, they're great. Not sure how close their course is to 174, but there may be a few useful practice problems on there
• Kimberly Brehm | Linear Algebra (Entire Course)
  • I haven't looked very far into this one, but according to a Sci 26, it has "similar content" although it's "in a bit of a different order"
• Dr. Trefor Bazett
  • Seems like it has plenty of good examples that relate to topics we're covering
• Wikipedia | Linear Algebra
  • Wikipedia can often be dense, technical, and often goes beyond what we will learn in 174, so it may not help a ton, however, it is written in the same language as 174, and may aid in clearing things up

Section 0 - Set Notation & Mappings

Set notation isn't super difficult, but this Wikipedia page provides a good reference point. Maps are essentially functions, if you want more, check out this Wikipedia page

Section 1 - Systems of Linear Equations

• MIT 18.06SC | 1. The Geometry of Linear Equations
• Dr. Trefor Bazett | What is a solution to a Linear System

Section 2 - Real Vector Spaces

• Professor Dave Explains | Understanding Real Vector Spaces
• Dr. Trefor Bazett | Vector Spaces
• Dr. Trefor Bazett | The Vector Space of Polynomials
• Kimberly Brehm | Vector Spaces - has lots of good practice questions
• 3Blue1Brown | Abstract Vector Spaces
• LibreTexts | 4.1: Definition of vector spaces
• Wikipedia | Vector space
• The Bright Side of Mathematics | Linear Algebra 2 | Vectors in ℝ²
• The Bright Side of Mathematics | Linear Algebra 5 | Vector Space ℝn

Section 3 - Vector Subspaces

• From MIT 18.06SC Lecture 5, Starts at 20:45 (We haven't hit the material before 20:45 yet)
• Khan Academy | Linear Subspaces
• Kimberly Brehm | Subspaces of a Vector Space
• Dr Trefor Bazett | Subspaces
• The Bright Side of Mathematics | Linear Algebra 6 | Linear Subspaces
• The Bright Side of Mathematics | Linear Algebra 7 | Examples for Subspaces

Section 4 - Linear Combinations & Span

• 3Blue1Brown | Linear combinations, span, and basis vectors
• Khan Academy | Linear Combinations and Span
• Kimberly Brehm | Linear Algebra 1.3.2 Linear Combinations
• Wikipedia | Linear Combination
• Wikipedia | Span
• How to determine if one vector is in the span of the other vectors
• The Bright Side of Mathematics | Linear Algebra 8 | Linear Span

Section 5 - Linear Independence

• Khan Academy| Linear Independence
• Exceprt from MIT 18.06SC Lecture 9 | Independence, Basis, and Dimension
  • Starting from 4:26 until 11:20 (ish)
  • This is unfortunalty not that good of an explanation as we haven't learnt enough about matrixes for what he is saying to really make sense. However, it may make more sense later.
• Kimberly Brehm | Linear Independence
• Kimberly Brehm | Special Methods of Determining Linear Independence
• Dr. Trefor Bazett | Linear Independence and Dependence - Geometrically
• The Bright Side of Mathematics | Linear Algebra 22 | Linear Independence (Definition)
• The Bright Side of Mathematics | Linear Algebra 23 | Linear Independence (Examples)

Section 6 - "Applications of Systems of Linear Equations"

This section boils down to asking: "how many solutions are there to: $w = \alpha_1 v_1 + \alpha_2v_2 +...\alpha_pv_p$?

The answer to this question is another question:

Is $w\in Span \left[ v_1,v_2,...,v_p \right]$?

No? Then there are no solutions.

Yes? Now, are $v_1,v_2,...,v_p$ dependant or independant?

Independant? There is only one solution

Dependant? There is infinitely many solutions.

Section 7 - Systems of Linear Equations

• Kimberly Brehm | Augmented Matrixes and Row Operations
• Kimberly Brehm | Row Reduction and Echelon Forms
• Khan Academy | Solving systems of equations with Matrixes
• Dr Trefor Bazett | Rewriting a Linear System using Matrix Notation
• Dr Trefor Bazett | Using Elementary Row Operations to Solve Systems of Linear Equations
• Dr Trefor Bazett | Using Elementary Row Operations to simplify a linear system
• Dr Trefor Bazett | Examples with 0, 1, and infinitely many solutions to linear systems
• Dr Trefor Bazett | Row Echelon Form and Reduced Row Echelon Form
• Dr Trefor Bazett | Back Substitution with infinitely many solutions
• Wikipedia | Elementary row operations
• The Bright Side of Mathematics | Linear Algebra 11 | Matrices
• The Bright Side of Mathematics | Linear Algebra 12 | Systems of Linear Equations

Section 8 & 9 - Generating Sets and Bases, Finite Dimensional Bases

A generating set for the RVS, $V$, is a set of vectors $v_1,v_2,...,v_p$ $\in V$ where $V$ is the span of ${v_1,v_2,...,v_p}$.

A basis for $V$ is a generating set for $V$ that is also linearly independant. And for a finite-dimensional basis, dim($V$) = minimum number of vectors required to form a basis for $V$.

• Kimberly Brehm | Linearly Independant sets and Bases
• Kimberly Brehm | The Spanning Set Theorem
• Kimberly Brehm | The Dimension of a Vector Space
• Kimberly Brehm | Subspaces of a Finite Dimensional Space
• Wikipedia | Generating Sets
• Wikipedia | Basis

Section 10 - Linear Transformations, Image and Kernel of a Linear Transformation

• Dr Trefor Bazett | Linear Transformations
• 3Blue1Brown | Linear transformations and matrices
• 3Blue1Brown | Inverse matrices, column space and null space
  • For our purposes, the collmun space is the image, and the null space is the kernel
• Khan Academy | Functions and linear transformations
• Khan Academy | Linear transformation examples
• Kimberly Brehm | Introduction to Linear Transformations
• The Bright Side of Mathematics | Linear Algebra 14 | Column Picture of the Matrix-Vector Product
• The Bright Side of Mathematics | Linear Algebra 15 | Row Picture
• The Bright Side of Mathematics | Linear Algebra 16 | Matrix Product
• The Bright Side of Mathematics | Linear Algebra 17 | Properties of the Matrix Product
• The Bright Side of Mathematics | Linear Algebra 18 | Linear Maps (Definition)
• The Bright Side of Mathematics | Linear Algebra 20 | Linear maps induce matrices
• The Bright Side of Mathematics | Linear Algebra 21 | Examples of Linear Maps

Section 11 - Image, Kernel, and rank-nullity

• MIT OpenCourseWare | 6. Column Space and Nullspace
• MIT OpenCourseWare | Vector Subspaces
• 3Blue1Brown | Inverse matrices, column space and null space
• Dr. Trefor Bazett | The Null Space & Column Space of a Matrix | Algebraically & Geometrically
• Dr. Trefor Bazett | Finding a Basis for the Nullspace or Column space of a matrix A
• Dr. Trefor Bazett | Finding a basis for Col(A) when A is not in REF form.
• Dr. Trefor Bazett | The Dimension of a Subspace | Definition + First Examples
• Dr. Trefor Bazett | Computing Dimension of Null Space & Column Space
• Dr. Trefor Bazett | The Dimension Theorem | Dim(Null(A)) + Dim(Col(A)) = n | Also, Rank!
• Kimberly Brehm | Linear Algebra 4.2.1 Null Spaces
• Kimberly Brehm | Linear Algebra 4.2.2 Column Spaces
• Wikipedia | row and column spaces
• Wikipedia | Rank-Nulity

Section 12 - Linear Transformations

• Dr. Trefor Bazett | Linear Transformations
• Dr. Trefor Bazett | Matrix Transformations are the same thing as Linear Transformations
• Dr. Trefor Bazett | Finding the Matrix of a Linear Transformation
• 3Blue1Brown | Three-dimensional linear transformations
• MIT OpenCourseWare | 30. Linear Transformations and Their Matrices
• MIT OpenCourseWare | Linear Transformations
• Wikipedia | Linear Map

Section 13 - Matrix Multiplication

• Dr. Trefor Bazett | The motivation and definition of Matrix Multiplication
• 3Blue1Brown | Matrix multiplication as composition
• Kimberly Brehm | Linear Algebra 2.1.2 Matrix Operations - Multiplication and Transpose
• Wikipedia | Matrix Multiplication

Section 14 - Invertability and Determinants

The Inverse Matrix

• MIT OpenCourseWare | 3. Multiplication and Inverse Matrices
• MIT OpenCourseWare | Inverse Matrices
• Dr. Trefor Bazett | You can "invert" matrices to solve equations...sometimes!
• Dr. Trefor Bazett | Find the Inverse of a Matrix
• Dr. Trefor Bazett | When does a matrix fail to be invertible? Also more "Big Theorem".
• Dr. Trefor Bazett | Visualizing Invertible Transformations (plus why we need one-to-one)
• Kimberly Brehm | Linear Algebra 2.2.1 The Inverse of a Matrix
• Kimberly Brehm | Linear Algebra 2.2.3 Elementary Matrices And An Algorithm for Finding A Inverse
• Kimberly Brehm | Linear Algebra 2.3.1 Characterizations of Invertible Matrices
• MIT OpenCourseWare | 20. Cramer's Rule, Inverse Matrix, and Volume
• Wikipedia | Invertible Matrix
• Wikipedia | Cramers Rule
• The Bright Side of Mathematics | Linear Algebra 29 | Identity and Inverses

The Determinant

• MIT OpenCourseWare | 18. Properties of Determinants
• MIT OpenCourseWare | Properties of Determinants
• MIT OpenCourseWare | 19. Determinant Formulas and Cofactors
• MIT OpenCourseWare | Determinants
• Dr. Trefor Bazett | Determinants - a "quick" computation to tell if a matrix is invertible
• Dr. Trefor Bazett | Determinants can be computed along any row or column - choose the easiest!
• 3Blue1Brown | The Determinant
• Kimberly Brehm | Linear Algebra 3.1.1 Introduction to Determinants
• Kimberly Brehm | Linear Algebra 3.1.2 Co-factor Expansion
• Kimberly Brehm | Linear Algebra 3.2.1 Properties of Determinants
• Wikipedia | Determinant

Some useful determinant tricks:

• Laplace Expansion
• Rule of Sarrus

Section 15 - Eigenstuffs and Diagonalization

Eigenvectors, values, and spaces

• MIT OpenCourseWare | 21. Eigenvalues and Eigenvectors
• MIT OpenCourseWare | Eigenvalues and Eigenvectors
• 3Blue1Brown | Eigenvectors and eigenvalues
• Dr. Trefor Bazett | What eigenvalues and eigenvectors mean geometrically
• Dr. Trefor Bazett | Using determinants to compute eigenvalues & eigenvectors
• Dr. Trefor Bazett | Example: Computing Eigenvalues and Eigenvectors
• Kimberly Brehm | Linear Algebra 5.1.1 Eigenvectors and Eigenvalues
• Kimberly Brehm | Linear Algebra 5.1.2 More About Eigenvectors and Eigenvalues
• Kimberly Brehm | Linear Algebra 5.2.1 Determinants and the IMT
• Kimberly Brehm | Linear Algebra 5.2.2 The Characteristic Equation
• Wikipedia | Eigenvalues and eigenvectors

Diagonalization

• MIT OpenCourseWare | 22. Diagonalization and Powers of A
• MIT OpenCourseWare | Powers of a Matrix
• Dr. Trefor Bazett | Diagonal Matrices are Freaking Awesome
• Dr. Trefor Bazett | How the Diagonalization Process Works
• Dr. Trefor Bazett | Compute large powers of a matrix via diagonalization
• Dr. Trefor Bazett | Full Example: Diagonalizing a Matrix
• Dr. Trefor Bazett | Visualizing Diagonalization & Eigenbases
• Wikipedia | Eigendecomposition(diagonalization) of a matrix