Basis and dimension

Overview

flowchart TD
    A(Basis, Span <br/> Dimensions) --> B[Matrix column span]
    B-->|echelon form: <br/>columns with pivots|C[ linearly independent. <br/>Original columns basis<br/>span the column space]

Figure 1: A flow chart of concepts in Basis and Span

Span

The span of two vectors $\overline v$ and $\overline w$ is the set of all linear combinations with $a*\overline v + b * \overline w$ with $a,b \in R$. So two operations are used: vector scaling and vector addition.

Basis

When are vectors linear independent?

Example

$s_1*v_1 + ...+ s_3*v_3 = 0$

$$\begin{pmatrix} -2 & 1 & 4 & |& 0 \\ 6 & 1 & -12 &|& 0 \\ 1 & 1 & -2 &| & 0 \end{pmatrix}$$

If we only have trivial solution to $s_1*v_1 + ...+ s_3*v_3 = 0$ than the vectors are linearly independent. So we solve with gaussian elimination and get the following echelon form:

$$\begin{pmatrix} -2 & 1 & 4 & |& 0 \\ 0 & 4 & 0 & | & 0 \\ 0 & 0 & 0 & | & 0 \end{pmatrix}$$

• we se that we have a free parameter t in the $s_3$ because there is no pivot.
• $s_3 = t$
• $s_2 = 0$
• $s_1 = 2t$

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