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In Exercises $7-10,$ let $W$ be the subspace spanned by the $\mathbf{u}^{\prime}$ 's, and write $\mathbf{y}$ as the sum of a vector in $W$ and a vector orthogonal to $W$$$\mathbf{y}=\left[\begin{array}{l}{1} \\ {3} \\ {5}\end{array}\right], \mathbf{u}_{1}=\left[\begin{array}{r}{1} \\ {3} \\ {-2}\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{l}{5} \\ {1} \\ {4}\end{array}\right]$$

$y=\left[\begin{array}{c}{\frac{10}{3}} \\ {\frac{2}{3}} \\ {\frac{8}{3}}\end{array}\right]+\left[\begin{array}{c}{\frac{-7}{3}} \\ {\frac{7}{3}} \\ {\frac{7}{3}}\end{array}\right]$

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 3

Orthogonal Projections

Vectors

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So you want to write the vector? Why? As a sum, um, a factor And w and a vector orthogonal to w or W is a span off the to use. I was So to do this when you look at, uh, the earth orginal projection of why onto this band of you wanted you two to get the vectors in W s o. The formula for that is a vector. No more protection. Why hat? We project onto one. It's a top product and we add a projection onto you too. So first, let's find this vector. Eso Let's find over dot products you have. Why dot you won Does he call to negative one plus four plus three. So we get six. Uh, you want a product with itself is just gonna be one plus one plus one, 43 So those first coefficient six over three, which gives us two, uh, for the next one. Why dot product with you too, is one plus 12 minus six, which gives us 30 minus six or seven. So you two dot product with itself one plus nine plus four. That gives us 14. It's our second coefficient is seven over 14 or one. So let's go ahead and multiplies coefficients in. So to find you one gives us 2 to 2 and 1/2 multiplied by a factor. You too gives us minus 1/2 three halfs and get up on so adding use together to minus 1/2 iss Very halfs. Four halfs and three half's gives us seven halfs and two minus one is So this is director. Why hat and so to find her other vector, we just need us attract. So we're going to take, uh, why Ann's a track. Why hat? So we get what we need to add. So why I had to get back to the vector y. So why minus why hat? So we have negative one minus the halfs. Four minus and a halfs and three minus one, which gives us negative five house more minus 7/2 sis 1/2 and three minus Want us to so we can write Why, as the son of this vector and why had

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