MATH 323 Lecture 13

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Brain dump on Vector Space

Review

1. span: $\mathrm {Span} (v_{1},\ldots ,v_{n})\subseteq V$
2. linear independence: $c_{1}\,v_{1}+\dots +c_{n}\,v_{n}=0$
3. if 1 & 2, then $v$'s define a basis ($w=c_{1}\,v_{1}+\dots +c_{n}\,v_{n}\quad \forall w\in V$
4. cardinality of basis is dimension
   • if $\{v_{i}\}_{i=1}^{n}$ and $\{w_{j}\}_{j=1}^{k}$ are bases for $V$, then $n=k$.
   • Note: $\dim V=0\iff V=\{0\}$

Bases of Subspace

For a vector space $V$ with dimension less than ∞, if $W\subset V$, then $\dim W<\dim V$.

Infinite Dimension

Let $P=\bigcup _{n=1}^{\infty }P_{n}$ represent the space of polynomials. Then $\dim P_{n}=|\{1,x,\ldots ,x^{n-1}\}|=n$ and $\dim P=\infty$

Column and Row Space

For a $m\times n$ matrix, $M=(v_{1},\ldots ,v_{n})$, where $v$'s are columns.

$\mathrm {Span} (v_{1},\ldots ,v_{n})\subseteq \mathrm {R} ^{m}$ is called the column space of $M$.

$\mathrm {Span} ({\vec {w}}:{\mbox{rows of }}M)\subseteq \mathrm {R} ^{n}$ is called row space of $M$.

Theorem 3.4.4

For vector space $V$ with dimension $\dim V=n>0$, then

1. no set of fewer than $n$ vectors can span $V$;
2. any subset of fewer than $n$ linearly independent vectors can be extended (i.e. vectors can be added) to form a basis for $V$; and
3. any spanning set containing more than $n$ vectors can be pared down (i.e. vectors can be removed) to form a basis for $V$.

Bases in Euclidean Space

${\vec {v}}_{1},\ldots ,{\vec {v}}_{n}\in \mathbb {R} ^{n}$ is a basis iff $\left|M\right|\neq 0$, where $M=\left({\vec {v}}_{1},\ldots ,{\vec {v}}_{n}\right)$ is a $n\times n$ matrix.

Standard Bases

${\hat {\imath }},{\hat {\jmath }},{\hat {k}}$ is standard basis in $\mathbb {R} ^{3}$.

A standard basis for $\mathbb {R} ^{n}$ is a set $\{{\vec {e}}_{1},\ldots ,{\vec {e}}_{n}\}$ such that $i$^th entry of ${\vec {e}}_{i}$ is 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} , and all other entries are 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 0}

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, b]} denote an ordered set representation, such 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 [a,b] \ne [b,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 [\vec{e}_1, \vec{e}_2]} will denote an ordered basis 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 \mathbb{R}^2} , for example.

Conversion of Bases

To Standard Basis

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 \vec{u}_1, \vec{u}_2 \in \mathbb{R}^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 \mathrm{Span}(\vec{u}_1, \vec{u}_2) = \mathbb{R}^2} , and 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 [\vec{u}_1, \vec{u}_2]} be a second basis (i.e. not standard).

We can use basis vectors to describe any point 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} = c_1 \, \vec{u}_1 + c_2 \, \vec{u}_2} in 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}^2} .

1. Given 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} = \left\langle x_1, x_2 \right\rangle} , find its coordinates 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 [\vec{u}_1, \vec{u}_2]}
2. Given 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 c_1 \, \vec{u}_1 + c_2 \, \vec{u}_2} , find its coordinates 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 [\vec{e}_1, \vec{e}_2]} .

Examples

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{u}_1 = \left\langle 3, 2 \right\rangle = 3 \vec{e}_1 + 2 \vec{e}_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 \vec{u}_2 = \left\langle 1, 1 \right\rangle = \vec{e}_1 + \vec{e}_2} .

Case 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 c_1 \, \vec{u}_1 + c_2 \, \vec{u}_2 = c_1 ( 3 \vec{e}_1 + 2 \vec{e}_2 ) + c_2 ( \vec{e}_1 + \vec{e}_2 ) = (3c_1 + c_2) \vec{e}_1 + (2c_1 + c_2) \vec{e}_2} .

Case 1: 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} = \left\langle 3c_1 + c_2, 2c_1 + c_2 \right\rangle = \begin{bmatrix}3&1\\2&1\end{bmatrix} \, \begin{bmatrix}c_1\\c_2\end{bmatrix}} .

Note: The 2×2 matrix 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 U = (\vec{u}_1, \vec{u}_2)} is called the transition matrix.

Assuming 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 |U| \ne 0} , 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{c} = U^{-1} \, \vec{x}}

To Arbitrary 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 [\vec{v}_1, \vec{v}_2] \to [\vec{u}_1, \vec{u}_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 \vec{x} = c_1 \, \vec{v}_1 + c_2 \, \vec{v}_2 = d_1 \, \vec{u}_1 + d_2 \, \vec{u}_2}

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 V = (\vec{v}_1, \vec{v}_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 U = (\vec{u}_1 \vec{u}_2)} . Then (assuming 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 V} 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 U} are nonsingular)

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} V \, \vec{c} &= U \, \vec{d} \\ \vec{d} &= U^{-1} \, V \, \vec{c} \end{align}}

The 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 S = U^{-1}\,V} is the tranformation matrix 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{v}_1, \vec{v}_2]} to 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{u}_1, \vec{u}_2]}

Summary

Cases for transformation matrix 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}

1. 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{v}_1, \vec{v}_2] \to [\vec{e}_1, \vec{e}_2]: S = V}
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 [\vec{v}_1, \vec{v}_2] \to [\vec{u}_1, \vec{u}_2]: S = U^{-1}\, V}
3. 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{e}_1, \vec{e}_2] \to [\vec{u}_1, \vec{u}_2]: S = U^{-1}}

Note 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 S} for standard 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 [\vec{e}_1, \vec{e}_2]} is the identity matrix, so 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 = U^{-1} I = U^{-1}}

Coordinate Vector

For vector space 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 V} with dimension 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 \dim V = n} , 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 E = [v_1, v_2, \ldots, v_n]} be the ordered basis 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 V} . Thus 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 v = c_1 \, v_1 + c_2 \, v_2 + \dots + c_n \, v_n \quad \forall v \in V} .

The vector 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{c} = \left\langle c_1, c_2, \ldots, c_n \right\rangle \in \mathbb{R}^n} is called the coordinate vector 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 V} 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 E} and is denoted 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{c} = [V]_E} .

Therefore, the previous section could be rewritten 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 [X]_F = S[X]_E\,\!}
where 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} 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 F} are ordered bases 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 U} 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 V} , respectively, 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 S = U^{-1}V}

Review

Maximum number of points = 50 + 7 pt. bonus question

Look over Determinants, null spaces, and solving linear systems of equations.