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In Exercises $3-6,$ verify that $\left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\}$ is an orthogonal set, and then find the orthogonal projection of $\mathbf{y}$ onto $\operatorname{Span}\left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\}$$$\mathbf{y}=\left[\begin{array}{r}{-1} \\ {4} \\ {3}\end{array}\right], \mathbf{u}_{1}=\left[\begin{array}{l}{1} \\ {1} \\ {0}\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{r}{-1} \\ {1} \\ {0}\end{array}\right]$$

$\left[\begin{array}{c}{-1} \\ {4} \\ {0}\end{array}\right]$

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 3

Orthogonal Projections

Vectors

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So first we need to check that you wanted me to or thought Gotham by taking dot product So you won. Uh, not You too. So we get zero minus one plus one and that zero So those are welcoming each other. So now we need to find the doctor. Call it white hat. Um, what is the orthogonal projection of? Why? Onto the span of you wanted me to. So we find this by projecting Why onto your bomb. So you do that by taking dot product of the two over dot product of you one with itself and not apply. Buy you one and add to that the same thing. But projecting onto you too. Eso Let's compute all the dot products off to the side. Why dot You won gives us minus 24 minus four. Close one. So negative 28 plus warm negative 27. You won dot product with itself a 16 close one plus one. So we'll get 18. Why dot product with you too get zero plus four plus one. So five and finally you to doubt product with itself. Zero plus one plus one. So we get to so our first coefficient is negative. 27 or her 18 which we can simplify two minus three halfs. And 2nd 1 you get five over, Chip. So now we just want to multiply these coefficients through to the vectors s o negative. Three half's times you won give us 12 over to just six. And then they gave 3/2 times negative. One three halfs. Negative. 3/2 times, one minus that halves and then multiply it by five laps. And to you, too, you get a zero moon five halfs, five halfs. Adding these together 606 three halves and 5/2 says eight halves before and 5/8 minus three. Half says two halfs more s. So that is our director weren't interested in

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