15.10: Espacio de función
- Page ID
- 86590
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)También podemos encontrar vectores base (Sección 15.9) para espacios vectoriales (Sección 15.2) distintos a\(\mathbb{R}^n\).
Dejar\(P_n\) ser el espacio vectorial de polinomios de orden\(n\) -ésimo en (-1, 1) con coeficientes reales (verificar\(P_2\) es un v.s. en casa).
Ejemplo\(\PageIndex{1}\)
\(P_{2}=\{\text { all quadratic polynomials }\}\). Vamos\(b_0(t)=1\),\(b_1(t)=t\),\(b_2(t)=t^2\).
\(\left\{b_{0}(t), b_{1}(t), b_{2}(t)\right\}\)span\(P_2\), es decir, puede escribir cualquiera\(f(t) \in P_2\) como
\[f(t)=\alpha_{0} b_{0}(t)+\alpha_{1} b_{1}(t)+\alpha_{2} b_{2}(t) \nonumber \]
para algunos\(\alpha_{i} \in \mathbb{R}\).
Nota
\(P_2\)es de 3 dimensiones.
\[f(t)=t^2−3t−4 \nonumber \]
Bases alternas
\[\left\{b_{0}(t), b_{1}(t), b_{2}(t)\right\}=\left\{1, t, \frac{1}{2}\left(3 t^{2}-1\right)\right\} \nonumber \]
escribir\(f(t)\) en términos de esta nueva base\(d_0(t)=b_0(t)\),\(d_1(t)=b_1(t)\),\(d_{2}(t)=\frac{3}{2} b_{2}(t)-\frac{1}{2} b_{0}(t)\).
\ [\ begin {array} {c}
f (t) =t^ {2} -3 t-4=4 b_ {0} (t) -3 b_ {1} (t) +b_ {2} (t)\\
f (t) =\ beta_ {0} d_ {0} (t) +\ beta_ {1} d_ {1} (t) +\ beta_ {2} d_ {2} (t) =\ beta_ {0} b_ {0} (t) +\ beta_ {1} b_ {1} (t) +\ beta_ {2}\ izquierda (\ frac {3} {2} b_ {2} (t) -\ frac {1} {2} b_ {0} (t)\ derecha)\
f (t) =\ beta_ {0} b_ {0} (t) +\ beta_ {1 } b_ {1} (t) +\ frac {3} {2}\ beta_ {2} b_ {2} (t)
\ end {array}\ nonumber\]
por lo
\ [\ begin {array} {c}
\ beta_ {0} -\ frac {1} {2} =4\\
\ beta_ {1} =-3\\
\ frac {3} {2}\ beta_ {2} =1
\ end {array}\ nonumber\]
entonces conseguimos
\[f(t)=4.5 d_{0}(t)-3 d_{1}(t)+\frac{2}{3} d_{2}(t) \nonumber \]
Ejemplo\(\PageIndex{2}\)
\(\left.e^{j \omega_{0} n t}\right|_{n=-\infty} ^{\infty}\)es una base para\(L^2([0,T])\),\(T=\frac{2 \pi}{\omega_0}\),\(f(t)=\sum_{n} C_{n} e^{j \omega_{0} n t}\).
Calculamos los coeficientes de expansión con
Fórmula de “cambio de base”:
\[C_{n}=\frac{1}{T} \int_{0}^{T}\left(f(t) e^{-\left(j \omega_{0} n t\right)}\right) d t \nonumber \]
Nota
Hay un número infinito de elementos en el conjunto de bases, eso significa que\(L^2([0,T])\) es infinito dimensional (¡da miedo!).
Los espacios infinito-dimensionales son difíciles de visualizar. Podemos manejar la intuición reconociendo que comparten muchas de las mismas propiedades matemáticas con espacios dimensionales finitos. Muchos conceptos se aplican a ambos (como “expansión de base”). Algunos no (el cambio de base no es una buena fórmula matricial).