MATH 414 Lecture 17

« previous | Friday, February 21, 2014 | next »

End Exam 1 content

Inner Products

• Standard Inner products on $\mathbb {R} ^{n}$, $\mathbb {C} ^{n}$, $L^{2}$, and $\ell ^{2}$
• 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 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(x) = a_0 + \sum_{k=1}^\infty a_k \, \cos{(k\,x)} + b_k \, \sin{(k\,x)} \in L^2[-\pi,\pi]} , 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 \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 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(x) = \sum_{k=-\infty}^\infty \alpha_k \, \mathrm{e}^{i \, k \, x} \in L^2[-\pi, \pi]} , 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 \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 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, g \in L^2[-\pi, \pi]} , we get

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 \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}}