MATH 323 Lecture 18

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Announcements

• Test 2 will be November 13
• Math Club meeting Monday, November 5, 19:00 BLOC 117 — Dr. Frank Sotile on hyperbolic soccerballs.
• Office hours this week are 14:20–15:20

Similarity

$L:V\to V$

Let $A_{L}$ be the transformation matrix for $L$.

1. if $B$ is the matrix representing $L$ w.r.t. $[{\vec {u}}_{1},{\vec {u}}_{2}]$
2. if $A$ is the matrix representing $L$ w.r.t. $[{\vec {e}}_{1},{\vec {e}}_{2}]$
3. $U$ is the transition matrix corresponding to the change of basis from $[{\vec {u}}_{1},{\vec {u}}_{2}]$ to $[{\vec {e}}_{1},{\vec {e}}_{2}]$

Then $B=U^{-1}\,A\,U$.

$A$ is similar to $B$ iff the above equation holds for some nonsingular $U$

Example

Consider $L({\vec {x}})=\left\langle 2x_{1},x_{1}+x_{2}\right\rangle$

$A_{L,[{\vec {e}}_{1},{\vec {e}}_{2}]}={\begin{pmatrix}2&0\\1&1\end{pmatrix}}$

Let $[{\vec {u}}_{1}=\left\langle 1,1\right\rangle ,{\vec {u}}_{2}=\left\langle -1,1\right\rangle ]$ be another basis

${\begin{aligned}L({\vec {u}}_{1})&=A{\vec {u}}_{1}={\begin{pmatrix}2&0\\1&1\end{pmatrix}}\,{\begin{pmatrix}1\\1\end{pmatrix}}={\begin{pmatrix}2\\2\end{pmatrix}}\\L({\vec {u}}_{2})&=A{\vec {u}}_{2}={\begin{pmatrix}2&0\\1&1\end{pmatrix}}\,{\begin{pmatrix}-1\\1\end{pmatrix}}={\begin{pmatrix}-2\\0\end{pmatrix}}\end{aligned}}$

Transition matrix from $[{\vec {e}}_{1},{\vec {e}}_{2}]$ to $[{\vec {u}}_{1},{\vec {u}}_{2}]$ is $U=({\vec {u}}_{1},{\vec {u}}_{2})={\begin{pmatrix}1&-1\\1&1\end{pmatrix}}$

Transition matrix from $[{\vec {u}}_{1},{\vec {u}}_{2}]$ to $[{\vec {e}}_{1},{\vec {e}}_{2}]$ is $U^{-1}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}$

${\begin{aligned}U^{-1}L({\vec {u}}_{1})&={\begin{pmatrix}2\\0\end{pmatrix}}&U^{-1}L({\vec {u}}_{2})&={\begin{pmatrix}-1\\1\end{pmatrix}}\end{aligned}}$

The matrix $B={\begin{pmatrix}2&-1\\0&1\end{pmatrix}}=U^{-1}\,A\,U$ represents $L$ w.r.t. $[{\vec {u}}_{1},{\vec {u}}_{2}]$

Theorem 4.3.1

Let $E=[v_{1},\ldots ,v_{n}]$ and $F=[w_{1},\ldots ,w_{n}]$ be two ordered bases for a vector space $V$, and let $L$ be a linear operator on $V$.

Let $S$ be the transition matrix representing the change from $F$ to $E$.

If $A$ is the matrix representing $L$ w.r.t. $E$ and $B$ is the matrix representing $L$ w.r.t. $F$, then $B=S^{-1}\,A\,S$.

Proof

${\vec {x}}\in \mathbb {R} ^{n}$, $v=x_{1}\,w_{1}+\dots +x_{n}\,w_{n}$, and ${\hat {x}}=\left\langle x_{1},\ldots ,x_{n}\right\rangle =[v]_{F}$.

Let ${\vec {y}}=S{\vec {x}}=[v]_{E}$, ${\vec {t}}=A{\vec {y}}=[L(v)]_{E}$, ${\vec {z}}=B{\vec {x}}=[L(v)]_{F}$, where $S$ is transition matrix from $F$ to $E$ and $S^{-1}$ is transition matrix from $E$ to $F$.

$S^{-1}{\vec {t}}={\vec {z}}$, $S^{-1}AS{\vec {x}}={\vec {z}}=B{\vec {x}}$

Finding new Bases

$A$ represents $L$ w.r.t $E=[v_{1},\ldots ,v_{n}]$

Suppose we haeve $w_{1}=S_{11}\,v_{1}+\dots +S_{n1}\,v_{n}$ through $w_{n}=S{1n}\,v_{1}+\dots +S_{nn}\,v_{n}$. Then $F=[w_{1},\ldots ,w_{n}]$ gives us a new basis.

Example

$D:P_{3}\to P_{2}={\frac {\mathrm {d} }{\mathrm {d} x}}$ find $B$ representing $D$ w.r.t. 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 [1,x,x^2]} and 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} representing 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 D} w.r.t. 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 [1, 2x, 4x^2-2]} .

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 \begin{align} D(1) &= 0 & D(x) &= 1 & D(x^2) &= 2x \\ B &= \begin{bmatrix}0&1&0\\0&0&2\\0&0&0\end{bmatrix} \\ D(1) &= 0 & D(2x) &= 2 & D(4x^2-2)&= 8x \\ A &= \begin{bmatrix}0&2&0\\0&0&4\\0&0&0\end{bmatrix} \end{align}}

Write $[1,2x,4x^{2}-2]$ as a linear combination 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 [1,x,x^2]} to find 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 S} :

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 \begin{align} 1 &= 1 \cdot 1 + 0 \cdot x + 0 \cdot x^2 \\ 2x &= 0 \cdot 1 + 2 \cdot x + 0 \cdot x^2 \\ 4x^2-2 &= -2 \cdot 1 + 0 \cdot x + 4 \cdot x^2\\ S &= \begin{bmatrix}1&0&-2\\0&2&0\\0&0&4\end{bmatrix} \\ S^{-1} &= \begin{bmatrix}1&0&\frac{1}{2}\\0&\frac{1}{2}&0\\0&0&\frac{1}{4}\end{bmatrix} \end{align}}

Thus it holds that 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 B = S^{-1} \, A \, S} .

Chapter 5: Orthogonality

Scalar Product

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},\vec{y} \in \mathbb{R}^n}

Scalar product (also called dot product) is defined 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 \begin{align} \vec{x} \cdot \vec{y} &= \sum_{i=1}^n x_i \, y_i = x_1 \, y_1 + \dots + x_n \, y_n \\ &= \vec{x}^T \vec{y} = (x_1, \ldots, x_n) \begin{pmatrix}y_1\\\vdots\\y_n\end{pmatrix} \end{align}}

Length

The length of an 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 n} -dimensional vector is given by

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 \left\| \vec{x} \right\| = \sqrt{\sum_{x=1}^n x_i^2}}

The distance between two vectors 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}} and 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{y}} can be found from 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}-\vec{y}\|}

Theorem 5.1.1

If 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}, \vec{y} \in \mathbb{R}^n} and 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 \theta} is the angle between them, then

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} \cdot \vec{y} = \| \vec{x} \| \| \vec{y} \| \cos{\theta}}