MATH 323 Lecture 14

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Conversion of Bases

Coordinates of ${\vec {x}}$ 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 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 = [\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:

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

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

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

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