Workbook 22

Chapter 1: Span

PDF

Objectives

Be able to determine the span of vectors in either a line, a plane, or all numbers

Prerequisites

Understand linear combinations of vectors. from Chapter 3: "Linear Combinations of Vectors" in Workbook 21
lin_independence (Missing reference)
Be able to identify and understand the fundamental subspaces: nullspace, rowspace, and columnspace. from Chapter 7: "Subspaces" in Workbook 21

Chapter 2: Determinants and Inverse Matrices

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Objectives

Be able to find the determinant and understand what different values of it mean

Video: The determinant | Chapter 6, Essence of linear algebra - YouTube
Video: Determinant of 3x3 Matrices, 2x2 Matrix, Precalculus Video Tutorial - YouTube
Video: 3 x 3 determinant | Matrix transformations | Linear Algebra | Khan Academy - YouTube
Video: Interpreting determinants in terms of area | Matrices | Precalculus | Khan Academy - YouTube
Video: Finding the determinant of a 2x2 matrix | Matrices | Precalculus | Khan Academy - YouTube
Video: 18. Properties of Determinants - YouTube
Video: Properties of Determinants - YouTube
Video: Determinants | MIT 18.02SC Multivariable Calculus, Fall 2010 - YouTube
Reference: Determinant - Wikipedia
Reference: Leibniz formula for determinants - Wikipedia
Reference: Laplace expansion - Wikipedia
Reference: 3Blue1Brown - The determinant

Be able to to find inverses of matrices, and use them for practical vector problems.

Prerequisites

Transform systems of equations into matrices. Understand Gaussian and Gauss-Jordan Elimination, Row Echelon Form, and Reduced Row Echelon Form. Identify free variables, dimension, basics of basis. from Chapter 4: "Systems of Equations as Matrices" in Workbook 21
Determine whether a set of vectors are independent or independent. Understand the basics of a basis and span. from Chapter 6: "Linear Independence and Dependence" in Workbook 21
Understand linear combinations of vectors. from Chapter 3: "Linear Combinations of Vectors" in Workbook 21

Chapter 3: Cross Product

PDF

Objectives

Understand the cross product of two vectors

Video: Cross products | Chapter 10, Essence of linear algebra - YouTube
Video: Cross Product of Two Vectors Explained! - YouTube
Video: 30.2 Cross Product - YouTube
Video: Dot vs. cross product | Physics | Khan Academy - YouTube
Video: Cross product introduction | Vectors and spaces | Linear Algebra | Khan Academy - YouTube
Video: Calculus 3 Lecture 11.4: The Cross Product - YouTube
Video: Right-hand rule for vector cross product - YouTube
Reference: Cross product - Wikipedia
Reference: Vector Calculus: Understanding the Cross Product – BetterExplained
Reference: Orthogonality (mathematics) - Wikipedia
Reference: Triple product - Wikipedia
Reference: 3Blue1Brown - Cross products

Prerequisites

Understand and multiply vectors using the dot product from Chapter 2: "The Dot Product" in Workbook 11
Be able to find the determinant and understand what different values of it mean from Chapter 2: "Determinants and Inverse Matrices" in Workbook 22
Be able to determine the span of vectors in either a line, a plane, or all numbers from Chapter 1: "Span" in Workbook 22

Chapter 4: Matrices as Transformations

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Downloads

Objectives

Understand the idea that matrices can be represented as transformations. Be able to rotate, reflect, and scale basis or points with matrices. Begin thinking about eigenvectors.

Prerequisites

Understand matrices, vector representations of matrices, dimensions of matrices, scalar matrix multiplication, and the zero matrix and identity matix. from Chapter 2: "Vectors and Matrices" in Workbook 21
Understand the 3 types of transformations from Chapter 1: "Congruence" in Workbook 05
Be able to to find inverses of matrices, and use them for practical vector problems. from Chapter 2: "Determinants and Inverse Matrices" in Workbook 22
Be able to find the determinant and understand what different values of it mean from Chapter 2: "Determinants and Inverse Matrices" in Workbook 22

Chapter 5: Eigenvectors and Eigenvalues

PDF

Downloads

Python script for creating and using eigenvalues and vectors.

Objectives

Understand how to compute eigenvalues and eigenvectors

Prerequisites

Understand a vector and its properties in depth from Chapter 5: "Vectors" in Workbook 06
Be able to find the determinant and understand what different values of it mean from Chapter 2: "Determinants and Inverse Matrices" in Workbook 22
Understand linear combinations of vectors. from Chapter 3: "Linear Combinations of Vectors" in Workbook 21
Understand and multiply vectors using the dot product from Chapter 2: "The Dot Product" in Workbook 11
lin_matrix_transformations (Missing reference)

Chapter 6: Projections

PDF

Downloads

Python file for calculating vector projections.

Objectives

Understand vector projections, what they create, and how to calculate them using matrices

Video: Introduction to projections | Matrix transformations | Linear Algebra | Khan Academy - YouTube
Video: Calculus 3 - Vector Projections & Orthogonal Components - YouTube
Video: Vector Projections | Vector Calculus #17 - YouTube
Video: Dot products and duality | Chapter 9, Essence of linear algebra - YouTube
Video: 15. Projections onto Subspaces - YouTube
Video: Projection into Subspaces - YouTube
Video: The Math behind (most) 3D games - Perspective Projection - YouTube
Video: Vector Projections - YouTube
Video: Vector Projections Are Just Shadows - YouTube
Reference: Projection (linear algebra) - Wikipedia
Reference: Vector projection - Wikipedia
Reference: 2.6: The Vector Projection of One Vector onto Another - Mathematics LibreTexts
Reference: Scalar projection - Wikipedia

Prerequisites

Understand a vector and its properties in depth from Chapter 5: "Vectors" in Workbook 06
Understand and multiply vectors using the dot product from Chapter 2: "The Dot Product" in Workbook 11
Define a unit vector from Chapter 5: "Vectors" in Workbook 06

Generated 2026-02-09T16:24.