MATH 414 Lecture 17

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End Exam 1 content

Inner Products

• Standard Inner products on 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} , 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{C}^n} , 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 L^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 \ell^2}
• 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 \left\langle Y, X \right\rangle = Y^T \, A \, X} an inner product (given a 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 A} )?
  • Positivity: 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\langle \vec{v}, \vec{v} \right\rangle > 0} for all nonzero 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}}
  • Homogeneity: 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\langle c \, \vec{u}, \vec{v} \right\rangle = c \, \left\langle \vec{u}, \vec{v} \right\rangle} for all 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{u}, \vec{v}} and scalars 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 \in \mathbb{C}} .
  • (Conjugate) symmetry: 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 \overline{ \left\langle \vec{u}, \vec{v} \right\rangle } = \left\langle \vec{v}, \vec{u} \right\rangle} for all 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{u}, \vec{v}}
  • Linearity: 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\langle \vec{u} + \vec{v}, \vec{w} \right\rangle = \left\langle \vec{u}, \vec{w} \right\rangle + \left\langle \vec{v}, \vec{w} \right\rangle}
• Angle between 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 \cos{\theta} = \frac{\vec{u} \cdot \vec{v}}{ \left\| \vec{u} \right\| \, \left\| \vec{v} \right\|}}
• Length of a 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 \left\| \vec{u} \right\| = \sqrt{ \left\langle \vec{u}, \vec{u} \right\rangle}}
• 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 d = \left\| \vec{u} - \vec{v} \right\|}

Fourier Series

Parseval's Equation

Real Version

If $f(x)=a_{0}+\sum _{k=1}^{\infty }a_{k}\,\cos {(k\,x)}+b_{k}\,\sin {(k\,x)}\in L^{2}[-\pi ,\pi ]$, then

${\frac {1}{\pi }}\int _{-\pi }^{\pi }\left|f(x)\right|^{2}\,\mathrm {d} x=2\left|a_{0}\right|^{2}+\sum _{k=1}^{\infty }\left|a_{k}\right|^{2}+\left|b_{k}\right|^{2}$

Complex Version

If $f(x)=\sum _{k=-\infty }^{\infty }\alpha _{k}\,\mathrm {e} ^{i\,k\,x}\in L^{2}[-\pi ,\pi ]$, then

${\frac {1}{2\pi }}\left\|f\right\|^{2}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left|f(x)\right|^{2}\,\mathrm {d} x=\sum _{k=-\infty }^{\infty }\left|\alpha _{k}\right|^{2}$

Moreover, for $f,g\in L^{2}[-\pi ,\pi ]$, we get

${\frac {1}{2\pi }}\,\left\langle f,g\right\rangle ={\frac {1}{2\pi }}\,\int _{-\pi }^{\pi }f(t)\,{\overline {g(t)}}\,\mathrm {d} t=\sum _{n=0}^{\infty }\alpha _{n}\,{\overline {\beta _{n}}}$