MATH 323 Lecture 14

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Conversion of Bases

Coordinates of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}} w.r.t. basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E=[v_1,\ldots,v_n]} are written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\vec{x}]_E} .

Conversion from basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} to basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} takes on following notation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\vec{x}]_F = S[\vec{x}]_E}

For $E=[{\vec {e}}_{1},\ldots ,{\vec {e}}_{n}]$ (standard basis), $V=[{\vec {v}}_{1},\ldots ,{\vec {v}}_{n}]$ , and $U=[{\vec {u}}_{1},\ldots ,{\vec {u}}_{n}]$

Transformation matrix $S$ takes following values:

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

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

$\mathrm {rank} (A)=\dim({\mbox{row space of }}A)$

Example

$A={\begin{bmatrix}1&-2&3\\2&-5&1\\1&-4&-7\end{bmatrix}}\longrightarrow {\begin{bmatrix}1&-2&3\\0&1&5\\0&0&0\end{bmatrix}}$

the rank of A is 2
the vectors $\left\langle 1,-2,3\right\rangle$ and $\left\langle 0,1,5\right\rangle$ are linearly independent
they form a basis for the row span of $A$

Solving Linear System of Equations

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

Theorem 3.6.3

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} be a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times n} matrix.

The linear system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\vec{x} = \vec{b}} is consistent for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{b} \in \mathbb{R}^n} iff the column vectors of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} span Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n} .
The system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\vec{x} = \vec{b}} has at moste one solution for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{b}\in\mathbb{R}^n} iff the column vectors of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} are linearly independent.