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# Linear Algebra Videos

## Linear Algebra Videos

**Linear Algebra 1: Solving Systems of Linear Equations #1**

Linear Algebra is the study of Linear Maps. In practice, this comes down to solving systems of linear equations, represented via Matrices. In general, the possible behaviour of such systems have three types of behaviour. In this video we study the most common case, when the system has a single solution.

** Linear Algebra 2: Solving Systems of Linear Equations - No Solutions #2**

In this video, we study the second type of system of linear equations, systems with no solutions. This is perhaps the easiest type of system to solve, as long as you know how to identify it. We go through a method of formally proving if a system has no solutions.

**Linear Algebra 3: Solving Systems of Linear Equations - Infinite Solutions #3**

In this video, we study the final type of systems of linear equations, systems with infinite solutions. In practice, it can be easy to confuse these with systems with no solutions, so it's important to know the difference between the two. These are the most difficult systems to get complete answers to, but here we show how to classify all of the infinite solutions to a system of linear equations.

**Linear Algebra 4: How to Multiply Matrices **

Matrix Multiplication, or Matrix Composition, is an important part of understanding Linear Algebra, but the formula is often hard to remember an apply. In this video, we try establish a method of multiplication that is a little more intuitive, and easier to remember than the explicit formula.

** Linear Algebra 5: Computing the Determinant of a Matrix**

The Determinant is a handy operation that assigns a number to a square matrix. This number tells us important information about the matrix. Specifically, we care whether the determinant is zero or not. For example, if it is zero, we know that the matrix is non-invertible, while if it is non-zero, it is invertible. Similarly, if it is zero, it tells us that the corresponding linear system of equations will not have a single solution.

**Linear Algebra 6: Computing the Inverse of a Matrix**

An inverse matrix is a linear map that "undoes" the original matrix, which is formally represented by their product being the identity matrix. In this video, we go over how to construct the inverse Matrix for a square matrix.

**Linear Algebra 7: Understanding Linear Independence of Vectors**

Linear Independence is one of the most important aspects of linear algebra, but is often difficult to fully understand. It is a formalisation of the idea of a collection of vectors "not being in the same direction", generalised to make sense in any dimension. Linear Independant vectors have many important properties, and are required for constructing things like bases. Here, we go over how to check if a collection of vectors are linearly independant.

**Linear Algebra 8: Properties of a Span of Vectors**

The span of a collection of vectors is the set of vectors you can get by adding up scaled copies of vectors in your collection. Intuitively, it's the set of points you can get to by only travelling in the directions of the vectors in your collection. In this video, we go over how to check if a vector exists in a span of a collection of vectors.

**Linear Algebra 9: What is a Basis?**

A basis for a subspace is a means of descriping that subspace, and is a generalisation of the standard basis vectors of R^n. This video relies on understanding Linear Independance, and the Span, so make sure you check out our videos on those first. A basis is the minimum amount of information we need to identify a subspace. Intuitively, a basis of a space is the smallest possible collection of vectors that span a space. However, basis are not unique, but we can occasionally find a basis we want to work with which simplifies our problem.

**Linear Algebra 10: Computing a Basis for the Image of a Matrix**

The Image of a matrix is a very fundamental subspace associated to a matrix. Intuitively it is the space of possible outputs of a matrix, or equivalently, the span of its columns. In this video, we go over how to construct a basis for the Image.

**Linear Algebra 11: Computing a Basis for the Kernel of a Matrix**

The Kernel, similar to the Image, is a fundamental subspace associated to a matrix. It can be thought of as the set of vectors that get "squished" to the zero vector. In this video, we go over how to construct a basis for the Kernel.

**Linear Algebra 12: Introduction to Eigenvalues, and how to find the Eigenvalues of a 2x2 matrix**

Eigenvalues and Eigenvectors of a matrix are a pair of a real number and a vector, such that applying our matrix to this eigenvector results in an output that is our eigenvector scaled by the eigenvalue. These are quite complicated objects, and in this video, we solve the simple case of Eigenvalues and Eigenvectors for a 2x2 matrix.

**Linear Algebra 12.5: Finding Eigenvalues**

In this video, we briefly derive the method of solving for eigenvalues, and justify the method involving the Characteristic Polynomial.

**Linear Algebra 13: Finding the Eigenvalues of a 3x3 matrix**

In this video, we establish some methods to solve for Eigenvalues of a 3x3 matrix, which is a more involved problem than with a 2x2 matrix.