Allan Moreno
02/12/2023 · Elementary School
6. Construya una base ortonormal para el subespacio vectorial de \( \mathbb{R}^{3} \) dado https://youtu.be/t7bgfo-U-aU?si=iOB 9vRLvuGz2OMV: https://voutu.be/V95aRhnEbew?si=vDuWJCIGMoclx-UR \( ] \) a. \( W=\{(x, y, z) \mid 2 x-y-z=0\} \) b. \( H=\{(x, y, z, t) \mid x+2 y+3 z-t=0\} \) c. \( T=\{(3 t, 4 t, 0) \mid t \in \mathbb{R}\} \)
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### a. \( W=\{(x, y, z) \mid 2x - y - z = 0\} \)
La base ortonormal para \( W \) es:
\[
\left\{\left(\frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}\right)\right\}
\]
### b. \( H=\{(x, y, z, t) \mid x + 2y + 3z - t = 0\} \)
La base ortonormal para \( H \) es:
\[
\left\{\left(\frac{1}{\sqrt{50}}, \frac{2}{\sqrt{50}}, \frac{3}{\sqrt{50}}, \frac{-6}{\sqrt{50}}\right)\right\}
\]
### c. \( T=\{(3t, 4t, 0) \mid t \in \mathbb{R}\} \)
La base ortonormal para \( T \) es:
\[
\left\{\left(\frac{3}{5}, \frac{4}{5}, 0\right)\right\}
\]
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