MATH 323 Lecture 14

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Conversion of Bases

Coordinates of $\overrightarrow{x}$ w.r.t. basis $E=[v_1,\dots ,v_n]$ are written as $[\overrightarrow{x}{]}_E$.

Conversion from basis $E$ to basis $F$ takes on following notation:

$[\overrightarrow{x}{]}_F=S[\overrightarrow{x}{]}_E$

For $E=[{\overrightarrow{e}}_1,\dots ,{\overrightarrow{e}}_n]$ (standard basis), $V=[{\overrightarrow{v}}_1,\dots ,{\overrightarrow{v}}_n]$, and $U=[{\overrightarrow{u}}_1,\dots ,{\overrightarrow{u}}_n]$

Transformation matrix $S$ takes following values:

1. $V\to E:V$
2. $V\to U:U^{-1}V$
3. $E\to U:U^{-1}$

Row Space and Column Space

Span of row vectors or column vectors of matrix.

Theorem 3.6.1: Two row equivalent matrices have the same row space.

Rank of a Matrix

$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(A)=\dim (\text{row space of }A)$

Example

$A=\left[\begin{array}{ccc}1& -2& 3\\ 2& -5& 1\\ 1& -4& -7\end{array}\right]⟶\left[\begin{array}{ccc}1& -2& 3\\ 0& 1& 5\\ 0& 0& 0\end{array}\right]$

• the rank of A is 2
• the vectors $⟨1,-2,3⟩$ and $⟨0,1,5⟩$ are linearly independent
• they form a basis for the row span of $A$

Solving Linear System of Equations

Theorem 3.6.2: The system $A\overrightarrow{x}=\overrightarrow{b}$ has a solution iff $\overrightarrow{b}$ is contained in the column space of $A$.

Theorem 3.6.3

Let $A$ be a $m\times n$ matrix.

• The linear system $A\overrightarrow{x}=\overrightarrow{b}$ is consistent for every $\overrightarrow{b}\in {\mathbb{ℝ}}^n$ iff the column vectors of $A$ span ${\mathbb{ℝ}}^n$.
• The system $A\overrightarrow{x}=\overrightarrow{b}$ has at moste one solution for every $\overrightarrow{b}\in {\mathbb{ℝ}}^n$ iff the column vectors of $A$ are linearly independent.