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elchatosarapage

Answered question

2021-11-23

Let

y = [\begin{matrix} 2 \\ 3 \end{matrix}] a n d u = [\begin{matrix} 4 \\ - 7 \end{matrix}]

Write y as the sum of two orthogonal vectors, one in Span{u} and one orthogonal to u.

Answer & Explanation

Muspee

Beginner2021-11-24Added 13 answers

Formula for orthogonal projection:

\hat{y} = \frac{y \cdot u}{u \cdot u} u

y \cdot u = (2, 3) \cdot (4, - 7) = 8 - 21 = - 13

u \cdot u = (4, - 7) \cdot (4, - 7) = 16 + 49 = 65

\hat{y} = \frac{- 13}{65} u = \frac{- 1}{5} [\begin{matrix} 4 \\ - 7 \end{matrix}] = [\begin{matrix} - 4 / 5 \\ 7 / 5 \end{matrix}]

\hat{y}

is the part that is in span u. Now calculate other part:

y - \hat{y} = [\begin{matrix} 2 \\ 3 \end{matrix}] - [\begin{matrix} - 4 / 5 \\ 7 / 5 \end{matrix}] = [\begin{matrix} 14 / 5 \\ 8 / 5 \end{matrix}]

y = [\begin{matrix} - 4 / 5 \\ 7 / 5 \end{matrix}] + [\begin{matrix} 14 / 5 \\ 8 / 5 \end{matrix}]

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