Linear Algebra

• Theory syllabus

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Matrices

Matrices play a central role in linear algebra. They can be used to compactly represent systems of linear equations, but they also represent linear mappings.

Before we discuss some of these interesting topics, let us first define what a matrix is and what kind of operations we can do with matrices.

Matrix

With $m,n\in \mathbb{N}$ a real-valued $(m,n)$ matrix $\mathbf{A}$ is an $m\cdot n$-tuple of elements $a_{ij},i=1,\dots ,m,j=1,\dots ,n$, which is ordered according to a rectangular scheme consisting of $m$ rows and $n$ columns:

$$\mathbf{A}=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{11}& a_{12}& \dots & a_{1n}\\ ⋮& ⋮& & ⋮\\ a_{m1}& a_{m2}& \dots & a_{mn}\end{array}\right],a_{ij}\in \mathbb{R}$$

By convention $(1,n)$-matrices are called rows and $(m,1)$-matrices are called columns.

Solving systems of linear equations

Vector Spaces

Linear independence

Basis and Rank

Linear Mappings

Affine Spaces