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## Course Resources

MATH 2010 -Linear Algebra

## Resource List by Course Content

### Chapter 1 - Introduction

### Chapter 2 – First-Order Differential Equations

### Chapter 3 - Mathematical Models and Numerical Methods Involving First-Order Equations

### Chapter 4 - Linear Second Order Equations

### Chapter 6 - Theory of Higher-Order Linear Differential Equations

### Chapter 7 - Laplace Transforms

### Chapter 8 - Series Solutions to Differential Equations

### Khan Academy Linear Algebra Links:

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learn about our college

Investing in yourself pays lifelong dividends

Helping students pay for college

Providing university transfer and in-demand career programs

Helping students succeed

Offering non-credit courses for both business and pleasure

Meeting the area's workforce needs

Enhancing Walters State through private support

Providing opportunities for alumni to reconnect

Making education convenient

Providing agribusiness, educational and commercial activities and events

Get in touch with us

Go Senators!

Looking for something specific?

Search our site alphabetically

We keep you connected

Search our team

Student learning and teaching effectiveness development

Information and educational technologies

Supplying the institution with quality printed materials

Serving students and college departments

Bringing international events to our students

Get the latest emergency communications

Providing resources for our staff

Bringing new experiences to our students and community

Requirements and college information

Keep tabs on important dates

Humanities

Business

Natural Science

Mathematics

MATH 2010 -Linear Algebra

The resources listed on this page are meant only as general instruction materials. Students are expected to contact their instructors regarding the specific requirements for their classes.

***All video resources below link to Khan Academy unless otherwise noted**

- Introduction to matrices : What a matrix is. How to add and subtract them.
- Matrix multiplication (part 1) : Multiplying two 2x2 matrices
- Matrix multiplication (part 2) : More on multiplying matrices
- Inverse Matrix (part 1) : Taking the inverse of a 2x2 matrix
- Inverting matrices (part 2) : Inverting a 3x3 matrix
- Inverting Matrices (part 3): Using Gauss-Jordan elimination to invert a 3x3 matrix
- Matrices to solve a system of equations: Using the inverse of a matrix to solve a system of equations
- Matrices to solve a vector combination problem
- Singular Matrices: When and why you can't invert a matrix
- 3-variable linear equations (part 1) : Visual intuition of a 3-variable linear equation
- Solving 3 Equations with 3 Unknowns (old)
- Linear Algebra: Introduction to Vectors
- Linear Algebra: Vector Examples: Visually understanding basic vector operations
- Linear Algebra: Parametric Representations of Lines
- Linear Combinations and Span
- Linear Algebra: Introduction to Linear Independence
- More on linear independence
- Span and Linear Independence Example
- Linear Subspaces: Introduction to linear subspaces of Rn
- Linear Algebra: Basis of a Subspace: Understanding the definition of a basis of a subspace
- Vector Dot Product and Vector Length
- Proving Vector Dot Product Properties
- Proof of the Cauchy-Schwarz Inequality
- Linear Algebra: Vector Triangle Inequality
- Defining the angle between vectors
- Defining a plane in R3 with a point and normal vector
- Linear Algebra: Cross Product Introduction
- Proof: Relationship between cross product and sin of angle
- Dot and Cross Product Comparison/Intuition
- Matrices: Reduced Row Echelon Form 1
- Matrices: Reduced Row Echelon Form 2
- Matrices: Reduced Row Echelon Form 3
- Matrix Vector Products
- Introduction to the Null Space of a Matrix
- Null Space 2: Calculating the null space of a matrix
- Null Space 3: Relation to Linear Independence
- Column Space of a Matrix : Introduction
- Null Space and Column Space Basis
- Visualizing a Column Space as a Plane in R3
- Proof: Any subspace basis has same number of elements
- Dimension of the Null Space or Nullity
- Dimension of the Column Space or Rank
- Showing relation between basis cols and pivot cols
- Showing that the candidate basis does span C(A)
- A more formal understanding of functions
- Vector Transformations: Introduction
- Linear Transformations: Introduction
- Matrix Vector Products as Linear Transformations
- Linear Transformations as Matrix Vector Products
- Image of a subset under a transformation
- im(T): Image of a Transformation
- Preimage of a set: Definition
- Preimage and Kernel Example
- Sums and Scalar Multiples of Linear Transformations
- More on Matrix Addition and Scalar Multiplication
- Linear Transformation Examples: Scaling and Reflections
- Linear Transformation Examples: Rotations in R2
- Rotation in R3 around the X-axis
- Unit Vectors: What unit vectors are and how to construct them
- Introduction to Projections
- Expressing a Projection on to a line as a Matrix Vector product
- Compositions of Linear Transformations 1
- Compositions of Linear Transformations 2
- Linear Algebra: Matrix Product Examples
- Matrix Product Associativity
- Distributive Property of Matrix Products
- Linear Algebra: Introduction to the inverse of a function
- Proof: Invertibility implies a unique solution to f(x)=y
- Surjective (onto) and Injective (one-to-one) functions
- Relating invertibility to being onto and one-to-one
- Determining whether a transformation is onto
- Linear Algebra: Exploring the solution set of Ax=b
- Linear Algebra: Matrix condition for one-to-one trans
- Linear Algebra: Simplifying conditions for invertibility
- Linear Algebra: Showing that Inverses are Linear
- Linear Algebra: Deriving a method for determining inverses
- Linear Algebra: Example of Finding Matrix Inverse
- Linear Algebra: Formula for 2x2 inverse
- Linear Algebra: 3x3 Determinant
- Linear Algebra: nxn Determinant
- Linear Algebra: Determinants along other rows/cols
- Linear Algebra: Rule of Sarrus of Determinants
- Linear Algebra: Determinant when row multiplied by scalar
- Linear Algebra: (correction) scalar multiplication of row
- Linear Algebra: Determinant when row is added
- Linear Algebra: Duplicate Row Determinant
- Linear Algebra: Determinant after row operations
- Linear Algebra: Upper Triangular Determinant
- Linear Algebra: Simpler 4x4 determinant
- Linear Algebra: Determinant and area of a parallelogram
- Linear Algebra: Determinant as Scaling Factor
- Linear Algebra: Transpose of a Matrix
- Linear Algebra: Determinant of Transpose
- Linear Algebra: Transpose of a Matrix Product
- Linear Algebra: Transposes of sums and inverses
- Linear Algebra: Transpose of a Vector
- Linear Algebra: Rowspace and Left Nullspace
- Lin Alg: Visualizations of Left Nullspace and Rowspace
- Linear Algebra: Orthogonal Complements
- Linear Algebra: Rank(A) = Rank(transpose of A)
- Linear Algebra: dim(V) + dim(orthogonoal complelent of V)=n
- Lin Alg: Representing vectors in Rn using subspace members
- Lin Alg: Orthogonal Complement of the Orthogonal Complement
- Lin Alg: Orthogonal Complement of the Nullspace
- Lin Alg: Unique rowspace solution to Ax=b
- Linear Alg: Rowspace Solution to Ax=b example
- Lin Alg: Showing that A-transpose x A is invertible
- Linear Algebra: Projections onto Subspaces
- Linear Alg: Visualizing a projection onto a plane
- Lin Alg: A Projection onto a Subspace is a Linear Transformation
- Linear Algebra: Subspace Projection Matrix Example
- Lin Alg: Another Example of a Projection Matrix
- Linear Alg: Projection is closest vector in subspace
- Linear Algebra: Least Squares Approximation
- Linear Algebra: Least Squares Examples
- Linear Algebra: Another Least Squares Example
- Linear Algebra: Coordinates with Respect to a Basis
- Linear Algebra: Change of Basis Matrix
- Lin Alg: Invertible Change of Basis Matrix
- Lin Alg: Transformation Matrix with Respect to a Basis
- Lin Alg: Alternate Basis Transformation Matrix Example
- Lin Alg: Alternate Basis Tranformation Matrix Example Part 2
- Lin Alg: Changing coordinate systems to help find a transformation matrix
- Linear Algebra: Introduction to Orthonormal Bases
- Linear Algebra: Coordinates with respect to orthonormal bases
- Lin Alg: Projections onto subspaces with orthonormal bases
- Lin Alg: Finding projection onto subspace with orthonormal basis example
- Lin Alg: Example using orthogonal change-of-basis matrix to find transformation matrix
- Lin Alg: Orthogonal matrices preserve angles and lengths
- Linear Algebra: The Gram-Schmidt Process
- Linear Algebra: Gram-Schmidt Process Example
- Linear Algebra: Gram-Schmidt example with 3 basis vectors
- Linear Algebra: Introduction to Eigenvalues and Eigenvectors
- Linear Algebra: Proof of formula for determining Eigenvalues
- Linear Algebra: Example solving for the eigenvalues of a 2x2 matrix
- Linear Algebra: Finding Eigenvectors and Eigenspaces example
- Linear Algebra: Eigenvalues of a 3x3 matrix
- Linear Algebra: Eigenvectors and Eigenspaces for a 3x3 matrix
- Linear Algebra: Showing that an eigenbasis makes for good coordinate systems
- Vector Triple Product Expansion (very optional)
- Normal vector from plane equation
- Point distance to plane
- Distance Between Planes

All Khan Academy content is available for free at www.khanacademy.org