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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$$$\mathbf{y}=\left[\begin{array}{l}{3} \\ {4} \\ {5} \\ {6}\end{array}\right], \mathbf{u}_{1}=\left[\begin{array}{r}{1} \\ {1} \\ {0} \\ {-1}\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{l}{1} \\ {0} \\ {1} \\ {1}\end{array}\right], \mathbf{u}_{3}=\left[\begin{array}{r}{0} \\ {-1} \\ {1} \\ {-1}\end{array}\right]$$

$y=\left[\begin{array}{l}{5} \\ {2} \\ {3} \\ {6}\end{array}\right]+\left[\begin{array}{c}{-2} \\ {2} \\ {2} \\ {0}\end{array}\right]$

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 3

Orthogonal Projections

Vectors

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University of Michigan - Ann Arbor

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So we want to write. Uh, why as a sum of vectors one a w and one orthogonal of W where w is just, uh this pan of all these use. So first, to find the one N w we need to project why they were on Waned of project. Why North, Ogden Lee, onto the span of you want you to endure three. So to do that, we'll take our projection formula and project. Why? Onto you won. You too on you three. So let's find all these DA products off to the side. Why? I thought you want iss three plus four. Plus they're a minus six, which gives us one. You want dot product with itself. Make it one plus one. Hello, zero plus one. So I get three. So those first coefficient is 1/3 Next. Uh, why that product with you too gives us hurry. Close zero most five plus six. So eight and six gives us 14 and you, too dot product with itself. Look it 1.0 close one plus one. So three again. So our second coefficient iss 14 over three. Finally, uh, why dot product with you three we get zero minus four. Close five minus six. Negative. 10 plus five gives us minus five and you three dot product with itself again, we get zero plus one plus one plus one. So three. So our last coefficient is negative. 5/3. So now we're going Thio multiplies coefficients into all the inspectors. So 1/3 time's you won gives us 1/3. 1/3 0 You could have 1/3. Ah, second victor, 14 3rd multiplied to you too. Gives us 14 3rd everywhere. Um, except a zero in the second place. And lastly, negative 5/3 multiplied into you three. So we have a zero 5/3 minus 5/3 and 5/3. So we're gonna add all these up. 1/3 plus 14 3rd ISS 15 over three, which is five 1/3 and 5/3. Gives us 6/3 or two 14 3rd Minus 5/3 is 9/3 or three. And lastly, 14 3rd plus 5/3 and 19 over three minus one over three. Gives us 18 over three or six. So our production vector that's in W is 5236 So we need to find the one that's 4000. All the w. So to do that, we're just going to take the vector we want to get, which is why subtract. Why? Hat? So we have 3456 minus that vector we just found, which is 5 to 36 So this gives us negative too. Two, 20 So therefore, um, we can rate why? As a song, Uh 5236 plus minus 22 to go.

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