10.1: Vectores de coordenadas
- Page ID
- 115126
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Dejar\(V\) ser un espacio de producto interior finito-dimensional con producto interno\(\inner{\cdot}{\cdot}\) y dimensión\(\dim(V)=n\). Entonces\(V\) tiene una base ortonormal\(e=(e_1,\ldots,e_n)\), y, según el Teorem9.4.6~\ ref {THM:parsevalsidentity}, cada uno\(v\in V\) puede escribirse como
\ begin {ecuation*} v =\ sum_ {i=1} ^n\ inner {v} {e_i} e_i.\ end {ecuación*}
Esto induce un mapa
\ begin {ecuación*}
\ begin {split}
[\,\ cdot\,] _e: V &\ to\ mathbb {F} ^n\
v &\ mapsto\ begin {bmatrix}
\ inner {v} {e_1}\\\ vdots\\\ interior {v} {e_n}\ end {bmatrix},
\ end {split}
\ end {ecuación*}
que mapea el vector\(v\in V\) al vector de\(n\times 1\) columna de sus coordenadas con respecto a la base\(e\). El vector de columna\([v]_e\) se denomina vector de coordenadas de\(v\) con respecto a la base\(e\).
Ejemplo\(\PageIndex{1}\):
Recordemos que el espacio vectorial\(\mathbb{R}_1[x]\) de polinomios sobre\(\mathbb{R}\) de grado como máximo 1 es un espacio interno de producto con producto interno definido por
\ begin {ecuación*}
\ interior {f} {g} =\ int_0^1 f (x) g (x) dx.
\ end {equation*}
Luego\(e=(1,\sqrt{3}(-1+2x))\) forma una base ortonormal para\(\mathbb{R}_1[x]\). El vector de coordenadas del polinomio\(p(x)=3x+2\in \mathbb{R}_1[x]\) es, p.
\[ [p(x)]_e= \frac{1}{2} \begin{bmatrix} 7 \\ \sqrt{3} \end{bmatrix}. \]
Obsérvese también que el mapa\([\,\cdot\,]_e\) es un isomorfismo (es decir, que es un mapa lineal inyectivo y surytivo) y que también es producto interno conservante. Denote el producto interno habitual en\(\mathbb{F}^n\) por
\ begin {ecuación*}
\ interior {x} {y} _ {\ mathbb {F} ^n} =\ suma_ {k=1} ^n x_k\ overline {y} _k.
\ final {ecuación*}
Entonces
\ begin {ecuación*}
\ inner {v} {w} _V =\ inner {[v] _e} {[w] _e} _ {\ mathbb {F} ^n},\ qquad\ texto {para todos\(v,w\in V\),}
\ end {ecuación*}
desde
\ begin {multline*}
\ interior {v} {w} _V =\ suma_ {i, j=1} ^n\ interior {\ interior {v} {e_i} e_i} {\ interior {w} {e_j} e_j}
=\ suma_ {i, j=1} ^n\ interior {v} {e_i}\ overline {\ interior {w} {e_j}\ interior {e_i} {e_j}\\
=\ suma_ {i, j=1} ^n\ interior {v} {e_i}\ overline {\ interior {w} {e_j}}\ delta_ {ij}
=\ suma_ {i=1 } ^n\ interior {v} {e_i}\ overline {\ inner {w} {e_i}} =\ interior {[v] _e} {[w] _e} _ _ {\ mathbb {F} ^n}.
\ end {multline*}
Es importante recordar que el mapa\([\,\cdot\,]_e\) depende de la elección de la base\(e=(e_1,\ldots,e_n)\).