Get the answer to "Let W be the subspace spanned by the..."

High School

Efan Halliday

Answered question

2021-10-29

Let W be the subspace spanned by the u’s, and write y as the sum of a vector in W and a vector orthogonal to W.
$y = [\begin{matrix} - 1 \\ 4 \\ 3 \end{matrix}], u_{1} = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}], u_{2} = [\begin{matrix} - 1 \\ 3 \\ - 2 \end{matrix}]$

Answer & Explanation

Jayden-James Duffy

Skilled2021-10-30Added 91 answers

$\hat{y} = \frac{y \cdot u_{1}}{u_{1} \cdot u_{1}} u_{1} + \frac{y \cdot u_{2}}{u_{2} \cdot u_{2}} u_{2}$ $= \frac{6}{3} [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] + \frac{7}{14} [\begin{matrix} - 1 \\ 3 \\ - 2 \end{matrix}]$
$\hat{y} = [\begin{matrix} \frac{3}{2} \\ \frac{7}{2} \\ 1 \end{matrix}]$
$y - \hat{y} = [\begin{matrix} - 1 \\ 4 \\ 3 \end{matrix}] - [\begin{matrix} \frac{3}{2} \\ \frac{7}{2} \\ 1 \end{matrix}] = [\begin{matrix} \frac{- 5}{2} \\ \frac{1}{2} \\ 2 \end{matrix}]$
Answer: $y = [\begin{matrix} \frac{3}{2} \\ \frac{7}{2} \\ 1 \end{matrix}] + [\begin{matrix} \frac{- 5}{2} \\ \frac{1}{2} \\ 2 \end{matrix}]$

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