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The prerequisite to the course Linear Algebra for Beginners: Open Doors to Great Careers 2.

Would you like to learn a mathematics subject that is crucial for many highdemand lucrative career fields such as:
If you're looking to gain a solid foundation in Linear Algebra, allowing you to study on your own schedule at a fraction of the cost it would take at a traditional university, to further your career goals, this online course is for you. If you're a working professional needing a refresher on linear algebra or a complete beginner who needs to learn Linear Algebra for the first time, this online course is for you.
Why you should take this online course: You need to refresh your knowledge of linear algebra for your career to earn a higher salary. You need to learn linear algebra because it is a required mathematical subject for your chosen career field such as computer science or electrical engineering. You intend to pursue a masters degree or PhD, and linear algebra is a required or recommended subject.
Why you should choose this instructor: I earned my PhD in Mathematics from the University of California, Riverside. I have extensive teaching experience: 6 years as a teaching assistant at University of California, Riverside, over two years as a faculty member at Western Governors University, #1 in secondary education by the National Council on Teacher Quality, and as a faculty member at Trident University International.
In this course, I cover the core concepts such as:
After taking this course, you will feel CAREFREE AND CONFIDENT. I will break it all down into bitesized nobrainer chunks. I explain each definition and go through each example STEP BY STEP so that you understand each topic clearly. I will also be AVAILABLE TO ANSWER ANY QUESTIONS you might have on the lecture material or any other questions you are struggling with.
Practice problems are provided for you, and detailed solutions are also provided to check your understanding.
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Section 1: Introduction  

Lecture 1 
Introduction Lecture
Preview

03:44  
Section 2: Solving Systems of Linear Equations  
Lecture 2  11:13  
In this lecture, we discuss Gaussian Elimination and examples of solving a system of linear equations. Students will learn about the three moves in Gaussian elimination. 

Lecture 3  18:15  
In this lecture, the row echelon form is introduced and many examples of Gaussian elimination are worked out. Students will learn how to apply Gaussian elimination to solve systems of linear equations. Students will also see the three different cases in regard to solutions to systems of equations. 

Lecture 4  1 page  
Practice problems for Gaussian elimination 

Lecture 5  1 page  
Solutions to Problem set Gaussian elimination 

Lecture 6  3 pages  
Detailed solutions are provided if needed. 

Lecture 7  11:13  
The augmented matrix is introduced and the elementary row operations are defined. Students will learn how to apply elementary row operations to an augmented matrix. 

Lecture 8  06:32  
An additional example of applying row operations is provided. 

Lecture 9  1 page  
Practice problems for Elementary Row Operations 

Lecture 10  1 page  
Solutions to Problem set Elementary Row Operations 

Lecture 11  4 pages  
Detailed solutions to the problem set Elementary Row Operations are provided, if needed. 

Section 3: Vectors  
Lecture 12  18:57  
The vector operations of addition, scalar multiplication, and matrix multiplication are introduced and the definition of linear combination is provided. Students will learn how to add two vectors, multiply a vector by a scalar, and multiply a vector by a matrix. The student will understand what linear combinations are and what weights are. 

Lecture 13  2 pages  
Practice problems for Vector Operations and Linear Combinations 

Lecture 14  1 page  
Solutions to Problem set Vector Operations and Linear Combinations 

Lecture 15  16:16  
In this lecture, the notion of span is introduced and it is shown how a system of equations can be rewritten as a matrix equation. 

Lecture 16  06:26  
The notion of linear independence is introduced. 

Lecture 17  11:02  
An example of determining linear independence is provided. Students will learn how to determine when a set of vectors is linearly independent. 

Lecture 18  04:36  
A second example of determining linear independence is provided. Students will learn how to determine when a set of vectors is linearly independent. 

Lecture 19  1 page  
Practice problems for Linear Independence 

Lecture 20  1 page  
Solutions to Problem set Linear Independence 

Lecture 21  2 pages  
Detailed solutions to problem set Linear Independence are provided, if needed. 

Section 4: Matrix Operations  
Lecture 22  07:12  
The matrix operations of addition and scalar multiplication are introduced. Students will learn how to add two matrices and how to multiply a matrix by a scalar. 

Lecture 23  09:18  
The matrix operation of multiplication is introduced. Students will learn how to multiply two matrices. 

Lecture 24  1 page  
Practice problems for Matrix Operations 

Lecture 25  1 page  
Solutions to Problem set Matrix Operations 

Lecture 26 
Review Request

1 page  
Section 5: Properties of Matrix Addition and Scalar Multiplication  
Lecture 27  13:13  
The properties of additive commutativity and associativity are introduced. The distributivity properties are also introduced. Students will learn how to apply the additive commutativity, additive associativity, and distributivity properties. 

Lecture 28  14:25  
The properties of additive and multiplicative identities, additive inverses, and multiplicative associativity and distributivity are introduced. Students will learn how to apply properties of additive and multiplicative identities, additive inverses, and multiplicative associativity and distributivity. 

Lecture 29  1 page  
Practice problems for Properties of Matrix Operations 

Lecture 30  1 page  
Solutions to Problem Set Properties of Matrix Operations 

Lecture 31  3 pages  
Detailed solutions to problem set Properties of Matrix Operations is provided, if needed. 

Lecture 32  06:42  
The definition of transpose of a matrix is provided, and properties of the transpose are introduced. Students will learn how to find the transpose of a matrix and how to apply properties of transposes. 

Lecture 33  1 page  
Practice problems for Transpose of a Matrix. 

Lecture 34  1 page  
Solutions to Problem Set Transpose of a Matrix 

Lecture 35  1 page  
Detailed solutions to Problem Set Transpose of a Matrix are provided, if needed. 

Section 6: The Inverse of a Matrix  
Lecture 36  05:30  
The definition of inverse matrix is introduced and the formula for the inverse of a 2 by 2 matrix is given. Students will learn what an inverse matrix is. 

Lecture 37  10:56  
The process of GaussJordan Elimination is explained. Students will learn how to find the inverse of a matrix using GaussJordan Elimination. 

Lecture 38  06:03  
An additional example of GaussJordan Elimination is provided. 

Lecture 39  1 page  
Practice problems for Inverse of a Matrix 

Lecture 40  1 page  
Solutions to Problem Set Inverse of a Matrix 

Lecture 41  5 pages  
Detailed solutions to Problem set Inverse of a Matrix are provided, if needed. 

Section 7: Determinants  
Lecture 42  02:34  
The formula for the determinant of a 2 by 2 matrix is introduced. Students will learn how to find the determinant of a 2 by 2 matrix. 

Lecture 43  07:18  
The process of Cofactor Expansion is explained. Students will learn how to apply cofactor expansion to find the determinant of a 3 by 3 matrix. 

Lecture 44  05:51  
Additional examples of cofactor expansion are provided. 

Lecture 45  1 page  
Practice problems for Determinants 

Lecture 46  1 page  
Solutions to Problem Set Determinants 

Lecture 47  2 pages  
Detailed solutions to Problem set Determinants are provided, if needed. 

Lecture 48 
Review Request

1 page  
Section 8: Properties of Determinants  
Lecture 49  11:07  
The determinant of a product of two matrices and the determinant of a scalar multiple of a matrix are discussed. Students will be able to find the determinant of a product of matrices and the determinant of a scalar multiple of a matrix. 

Lecture 50  07:26  
The relationship between determinants and invertibility is explained. Students will be able to determine when a matrix is invertible by examining the determinant. 

Lecture 51  03:35  
The determinant of the transpose of a matrix is discussed. Students will learn how to find the determinant of the transpose of a matrix. 

Lecture 52  1 page  
Practice problems for Properties of Determinants 

Lecture 53  1 page  
Solutions to Problem set Properties of Determinants 

Lecture 54  2 pages  
Detailed solutions to Problem set Properties of Determinants are provided, if needed. 

Section 9: Vector Spaces  
Lecture 55  07:22  
The definition of vector space is explained. Students will learn what a vector space is. 

Lecture 56  13:43  
A proof is given to show that R2 is a vector space. Students will learn how to prove a set is a vector space. 

Lecture 57  12:18  
The proof of the claim that R2 is a vector space is continued. 

Lecture 58  16:46  
An additional example of a vector space is provided. 

Lecture 59  04:03  
The proof of the first five properties of a vector space applied to P2 is continued. 

Lecture 60  1 page  
Practice problems for Vector Spaces. 

Lecture 61  4 pages  
Solutions to Problem Set Vector Spaces 

Lecture 62  06:09  
Examples of sets that are not vector spaces are discussed. Students will learn how to detect when a set is a not a vector space. 

Lecture 63  1 page  
Practice problems for Sets that are not vector spaces. 

Lecture 64  2 pages  
Solutions to Problem set Sets that are not vector spaces. 

Section 10: Subspaces  
Lecture 65  09:55  
The definition of a subspace is provided, and the three subspace properties are outlined. An example of a proof showing that a subset of R2 is a subspace is demonstrated. Students will learn how to show that a subset of a vector space is a subspace. 

Lecture 66  03:38  
The definition of a trivial subspace and of a nontrivial subspace are provided. 

Lecture 67  05:17  
An example of a subspace of M(2,2), the set of all 2 by 2 matrices, is provided. 

Lecture 68  1 page  
Practice problems for Subspaces 

Lecture 69  2 pages  
Solutions to Problem Set Subspaces 

Lecture 70  09:13  
An example of a subset of R2 that is not a subspace of R2 is discussed. Students will learn how to show that a subset of a vector space is not a subspace of that vector space. 

Lecture 71  04:25  
An additional example of a subset that is not a subspace is discussed. 

Lecture 72  1 page  
Practice problems for Subsets that are not subspaces. 

Lecture 73  1 page  
Solutions to Problem Set Subsets that are not subspaces. 

Section 11: Span and Linear Independence  
Lecture 74  15:26  
Students will learn how to show that a subset of a vector space spans the vector space and how to show that a subset of a vector space does not span the vector space. 

Lecture 75  08:24  
Students will learn what the span of a subset of a vector space is. 

Lecture 76  09:35  
Students will learn what it means for a subset of a vector space to be linearly independent. 

Lecture 77  13:44  
Students will learn how to determine if a subset of a vector space is linearly independent or dependent. 

Lecture 78  1 page  
Practice problems for span and linear independence. 

Lecture 79  3 pages  
Solutions to Problem Set: Span and Linear Independence 

Section 12: Basis and Dimension  
Lecture 80  16:07  
Students will learn how to show that a subset of a vector space is a basis for the vector space. 

Lecture 81  09:52  
Students will learn how to find the dimension of a vector space. 

Lecture 82  1 page  
Practice problems for basis and dimension. 

Lecture 83  2 pages  
Solutions to Problem Set: Basis and Dimension 

Lecture 84  03:26  
Students will learn what the coordinates of a vector relative to a basis are and what the coordinate matrix of a vector relative to a basis is. 

Lecture 85  09:23  
Students will learn what change of basis means and what a transition matrix from one basis to another is. 

Lecture 86  13:02  
Students will learn how to find transition matrices from one basis to another. 

Lecture 87  1 page  
Practice problems for coordinates and change of basis. 

Lecture 88  2 pages  
Solutions to Problem Set: Coordinates and Change of Basis 
Hi there! My name is Richard Han. I earned my PhD in Mathematics from the University of California, Riverside. I have extensive teaching experience: 6 years as a teaching assistant at University of California, Riverside, over two years as a faculty member at Western Governors University, #1 in secondary education by the National Council on Teacher Quality, and as a faculty member at Trident University International. My expertise includes calculus and linear algebra. I am an instructor on Udemy for the courses Philosophy of Language: Solidify Critical Thinking Skills and Linear Algebra for Beginners: Open Doors to Great Careers.