Linear Algebra

About Course

In this course, We will explain the essentials of Linear Algebra (LA) and everything that you need to understand the basics of linear algebra. We will cover content such as:

Matrices and Linear System of Equations
Gaussian Elimination
Reduced Echelon Form and RREF
Matrix Algebra
Special Matrices, Diagonal Matrices, and Inverse Matrices
Inverse Matrices and The Inverses of Transposed Matrices
Determinants and computing the Determinant
Much more!

By the end of this course, you should very comfortable with linear algebra and be able to follow throw any Math which uses the (LA) notation.

You will also get answers to any questions on any (LA) questions that you might have for life.

Who this course is for:

Computer Science Students Who Want To Learn More
Students Who Want To Learn The (LA) For Machine Learning and Deep Learning
Anyone Who Is Interested In Math And Wants To Study (LA)

Course Content

Linear Algebra

What’s the big idea of Linear Algebra?

00:00
What is a Solution to a Linear System?

00:00
Visualizing Solutions to Linear Systems

00:00
Rewriting a Linear System using Matrix Notation

00:00
Using Elementary Row Operations to Solve Systems of Linear Equations

00:00
Using Elementary Row Operations to simplify a linear system

00:00
Examples with 0, 1, and infinitely many solutions to linear systems

00:00
Row Echelon Form and Reduced Row Echelon Form

00:00
Back Substitution with infinitely many solutions

00:00
What is a vector?

00:00
Introducing Linear Combinations & Span

00:00
How to determine if one vector is in the span of other vectors?

00:00
Matrix-Vector Multiplication and the equation Ax=b

00:00
Matrix-Vector Multiplication Example

00:00
Proving Algebraic Rules in Linear Algebra — Ex: A(b+c) = Ab +Ac

00:00
The Big Theorem, Part I

00:00
Writing solutions to Ax=b in vector form

00:00
Geometric View on Solutions to Ax=b and Ax=0.

00:00
Three nice properties of homogeneous systems of linear equations

00:00
Linear Dependence and Independence – Geometrically

00:00
Determining Linear Independence vs Linear Dependence

00:00
Making a Math Concept Map | Ex: Linear Independence

00:00
Transformations and Matrix Transformations

00:00
Three examples of Matrix Transformations

00:00
Linear Transformations

00:00
Are Matrix Transformations and Linear Transformation the same? Part I

00:00
Every vector is a linear combination of the same n simple vectors!

00:00
Matrix Transformations are the same thing as Linear Transformations

00:00
Finding the Matrix of a Linear Transformation

00:00
One-to-one, Onto, and the Big Theorem Part II

00:00
The motivation and definition of Matrix Multiplication

00:00
Computing matrix multiplication

00:00
Visualizing Composition of Linear Transformations

00:00
Elementary Matrices

00:00
You can “invert” matrices to solve equations…sometimes!

00:00
Finding inverses to 2×2 matrices is easy!

00:00
Find the Inverse of a Matrix

00:00
When does a matrix fail to be invertible? Also more “Big Theorem”.

00:00
Visualizing Invertible Transformations (plus why we need one-to-one)

00:00
Invertible Matrices correspond with Invertible Transformations **proof**

00:00
Determinants – a “quick” computation to tell if a matrix is invertible

00:00
Determinants can be computed along any row or column – choose the easiest!

00:00
Vector Spaces | Definition & Examples

00:00
The Vector Space of Polynomials: Span, Linear Independence, and Basis

00:00
Subspaces are the Natural Subsets of Linear Algebra | Definition + First Examples

00:00
The Span is a Subspace | Proof + Visualization

00:00
The Null Space & Column Space of a Matrix | Algebraically & Geometrically

00:00
The Basis of a Subspace

00:00
Finding a Basis for the Nullspace or Column space of a matrix A

00:00
Finding a basis for Col(A) when A is not in REF form.

00:00
Coordinate Systems From Non-Standard Bases | Definitions + Visualization

00:00
Writing Vectors in a New Coordinate System **Example**

00:00
What Exactly are Grid Lines in Coordinate Systems?

00:00
The Dimension of a Subspace | Definition + First Examples

00:00
Computing Dimension of Null Space & Column Space

00:00
The Dimension Theorem | Dim(Null(A)) + Dim(Col(A)) = n | Also, Rank!

00:00
Changing Between Two Bases | Derivation + Example

00:00
Visualizing Change Of Basis Dynamically

00:00
Example: Writing a vector in a new basis

00:00
What eigenvalues and eigenvectors mean geometrically

00:00
Using determinants to compute eigenvalues & eigenvectors

00:00
Example: Computing Eigenvalues and Eigenvectors

00:00
A range of possibilities for eigenvalues and eigenvectors

00:00
Diagonal Matrices are Freaking Awesome

00:00
How the Diagonalization Process Works

00:00
Compute large powers of a matrix via diagonalization

00:00
Full Example: Diagonalizing a Matrix

00:00
COMPLEX Eigenvalues, Eigenvectors & Diagonalization **full example**

00:00
Visualizing Diagonalization & Eigenbases

00:00
Similar matrices have similar properties

00:00
The Similarity Relationship Represents a Change of Basis

00:00
Dot Products and Length

00:00
Distance, Angles, Orthogonality and Pythagoras for vectors

00:00
Orthogonal bases are easy to work with!

00:00
Orthogonal Decomposition Theorem Part 1: Defining the Orthogonal Compliment

00:00
The geometric view on orthogonal projections

00:00
Orthogonal Decomposition Theorem Part II

00:00
Proving that orthogonal projections are a form of minimization

00:00
Using Gram-Schmidt to orthogonalize a basis

00:00
Full example: using Gram-Schmidt

00:00
Least Squares Approximations

00:00
Reducing the Least Squares Approximation to solving a system

00:00

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About Course

In this course, We will explain the essentials of Linear Algebra (LA) and everything that you need to understand the basics of linear algebra. We will cover content such as:

Who this course is for:

What Will You Learn?

Course Content

Linear Algebra

What’s the big idea of Linear Algebra?

What is a Solution to a Linear System?

Visualizing Solutions to Linear Systems

Rewriting a Linear System using Matrix Notation

Using Elementary Row Operations to Solve Systems of Linear Equations

Using Elementary Row Operations to simplify a linear system

Examples with 0, 1, and infinitely many solutions to linear systems

Row Echelon Form and Reduced Row Echelon Form

Back Substitution with infinitely many solutions

What is a vector?

Introducing Linear Combinations & Span

How to determine if one vector is in the span of other vectors?

Matrix-Vector Multiplication and the equation Ax=b

Matrix-Vector Multiplication Example

Proving Algebraic Rules in Linear Algebra — Ex: A(b+c) = Ab +Ac

The Big Theorem, Part I

Writing solutions to Ax=b in vector form

Geometric View on Solutions to Ax=b and Ax=0.

Three nice properties of homogeneous systems of linear equations

Linear Dependence and Independence – Geometrically

Determining Linear Independence vs Linear Dependence

Making a Math Concept Map | Ex: Linear Independence

Transformations and Matrix Transformations

Three examples of Matrix Transformations

Linear Transformations

Are Matrix Transformations and Linear Transformation the same? Part I

Every vector is a linear combination of the same n simple vectors!

Matrix Transformations are the same thing as Linear Transformations

Finding the Matrix of a Linear Transformation

One-to-one, Onto, and the Big Theorem Part II

The motivation and definition of Matrix Multiplication

Computing matrix multiplication

Visualizing Composition of Linear Transformations

Elementary Matrices

You can “invert” matrices to solve equations…sometimes!

Finding inverses to 2×2 matrices is easy!

Find the Inverse of a Matrix

When does a matrix fail to be invertible? Also more “Big Theorem”.

Visualizing Invertible Transformations (plus why we need one-to-one)

Invertible Matrices correspond with Invertible Transformations **proof**

Determinants – a “quick” computation to tell if a matrix is invertible

Determinants can be computed along any row or column – choose the easiest!

Vector Spaces | Definition & Examples

The Vector Space of Polynomials: Span, Linear Independence, and Basis

Subspaces are the Natural Subsets of Linear Algebra | Definition + First Examples

The Span is a Subspace | Proof + Visualization

The Null Space & Column Space of a Matrix | Algebraically & Geometrically

The Basis of a Subspace

Finding a Basis for the Nullspace or Column space of a matrix A

Finding a basis for Col(A) when A is not in REF form.

Coordinate Systems From Non-Standard Bases | Definitions + Visualization

Writing Vectors in a New Coordinate System **Example**

What Exactly are Grid Lines in Coordinate Systems?

The Dimension of a Subspace | Definition + First Examples

Computing Dimension of Null Space & Column Space

The Dimension Theorem | Dim(Null(A)) + Dim(Col(A)) = n | Also, Rank!

Changing Between Two Bases | Derivation + Example

Visualizing Change Of Basis Dynamically

Example: Writing a vector in a new basis

What eigenvalues and eigenvectors mean geometrically

Using determinants to compute eigenvalues & eigenvectors

Example: Computing Eigenvalues and Eigenvectors

A range of possibilities for eigenvalues and eigenvectors

Diagonal Matrices are Freaking Awesome

How the Diagonalization Process Works

Compute large powers of a matrix via diagonalization

Full Example: Diagonalizing a Matrix

COMPLEX Eigenvalues, Eigenvectors & Diagonalization **full example**

Visualizing Diagonalization & Eigenbases

Similar matrices have similar properties

The Similarity Relationship Represents a Change of Basis

Dot Products and Length

Distance, Angles, Orthogonality and Pythagoras for vectors

Orthogonal bases are easy to work with!

Invertible Matrices correspond with Invertible Transformations proof

Writing Vectors in a New Coordinate System Example

COMPLEX Eigenvalues, Eigenvectors & Diagonalization full example