Linear Algebra for Beginners: Open Doors to Great Careers

Students will learn how to find transition matrices from one basis to another.

Students will learn what change of basis means and what a transition matrix from one basis to another is.

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.

Students will learn how to find the dimension of a vector space.

Students will learn how to show that a subset of a vector space is a basis for the vector space.

Students will learn how to determine if a subset of a vector space is linearly independent or dependent.

Students will learn what it means for a subset of a vector space to be linearly independent.

Students will learn what the span of a subset of a vector space is.

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.

An additional example of a subset that is not a subspace is discussed.

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.

An example of a subspace of M(2,2), the set of all 2 by 2 matrices, is provided.

The definition of a trivial subspace and of a nontrivial subspace are provided.

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.

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.

The proof of the first five properties of a vector space applied to P2 is continued.

An additional example of a vector space is provided.

The proof of the claim that R2 is a vector space is continued.

A proof is given to show that R2 is a vector space. Students will learn how to prove a set is a vector space.

The definition of vector space is explained. Students will learn what a vector space is.

The determinant of the transpose of a matrix is discussed. Students will learn how to find the determinant of the transpose of a matrix.

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

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.

Additional examples of cofactor expansion are provided.

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.

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.

An additional example of Gauss-Jordan Elimination is provided.

The process of Gauss-Jordan Elimination is explained. Students will learn how to find the inverse of a matrix using Gauss-Jordan Elimination.

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.

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.

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.

The properties of additive commutativity and associativity are introduced. The distributivity properties are also introduced.

The matrix operation of multiplication is introduced. Students will learn how to multiply two matrices.

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.

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

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

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.

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.

An additional example of applying row operations is provided.

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.

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.

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.

An introduction to the course is provided.