# Linear Algebra

Linear algebra is the part of mathematics concerning vectors, vector spaces and linear mappings between such spaces.

### System of Linear Equations

What is a system of linear equations? How many solutions do they have and how can you solve them? In this video you will learn the answer to all these questions.

### Vectors and Linear Combinations

In this video you will learn about vector equations. You will learn how to add vectors in R^n, do scalar multiplications and how to check if a vector is a linear combination of other vectors.

### Solution Sets of Linear Equations

In this video you will learn about the solution sets of linear equations. You will learn how the row echelon form of a matrix is connected to the solution sets.

### Matrix Transformations

In this video you will learn about matrix transformations. You will learn how to write a function of vectors as a matrix transformation.

### Matrix Multiplication

How do you multiply matrices? This video will learn you what linear transformations are and what a composition of multiple linear transformations is. This is the first step in understanding multiplication of matrices.

### Inverse Matrices

In this video you will learn about the inverse of a matrix. You will learn when a matrix is invertible and how to find the inverse of a 2x2 matrix.

### Linear Subspaces

In this video you will learn about linear subspaces. You will learn about the superposition principle and will see examples of important linear subspaces.

### Dimensions

In this video you will learn about spaces with higher dimensions. You will learn how to describe them and how many parameters you need to do that.

### Determinants

In this video you will learn about the determinant of a matrix. You will learn how to calculate the determinant of every square matrix.

### Cramer's rule

In this video you are going to see how determinants can be used to solve a system of n linear equations in n variables if the solution is unique. This technique is called Cramer's rule.

### Inner Product and Orthogonality

When are vectors orthogonal? In this video you will learn about the inner product of vectors. With the inner product you can determine if vectors are orthogonal. You will also learn important properties of inner products.

### Orthogonal Projections

In this video you will learn how to find the orthogonal projection of a vector onto a plane in three-dimensional space.

### The Gram-Schmidt Process

Does an orthogonal basis always exist? And how can we find such a basis for a given subspace? These are the questions that are tackled in this video. This has everything to do with the Gram-Schmidt process.

### Least-squares Problems

Many problems in engineering fall in the area of data fitting. You will learn how to use the least-squares method to find the best fit of data to a model.

### Eigenvectors and Eigenvalues

In this video you will learn what eigenvectors and eigenvalues of matrices are. You will see examples of how you can verify if a vector is an eigenvector and a scalar is an eigenvalue of a matrix.

### Diagonalisable Matrices

In this video you will learn that there exists a "magic trick" to make matrix computations really easy! Remember diagonal matrices, they are easy to work with. You will learn that some non-diagonal matrices can be diagonalised to make computations easier.

### Complex Eigenvalues

In this video you will learn how to find the eigenvalues and eigenvectors of a matrix, even if the characteristic equation does not have any real solutions.

### Systems of Linear Differential Equations

In this video you will learn about differential equations and how linear algebra can help in solving differential equations.

### Symmetric Matrices

Did you know that every symmetric matrix is orthogonally diagonalisable? In this video you will learn more about it.

In this video you will learn about quadratic forms. In addition to their interesting mathematical properties quadratic forms are important because they arise in many areas of application such as economics, statistics and physics.

### True/False Questions Part I

How do you answer a True/False question? Where do you start and which steps do you follow? In this video you will learn how to do this using the KISS method.

### True/False Questions Part II

In this video you will learn how to find a counterexample for a True/False question, and how to write down your answer on the exam.