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In Exercises $9-12,$ find (a) the orthogonal projection of $\mathbf{b}$ onto Col $A$ and $(b)$ a least-squares solution of $A \mathbf{x}=\mathbf{b} .$$$A=\left[\begin{array}{rrr}{1} & {1} & {0} \\ {1} & {0} & {-1} \\ {0} & {1} & {1} \\ {-1} & {1} & {-1}\end{array}\right], \mathbf{b}=\left[\begin{array}{l}{2} \\ {5} \\ {6} \\ {6}\end{array}\right]$$

(a) $\hat{b}=\frac{1}{3} a_{1}+\frac{14}{3} a_{2}+\frac{-5}{3} a_{3}=\frac{1}{3}\left[\begin{array}{c}{1} \\ {1} \\ {0} \\ {0} \\ {-1}\end{array}\right]+\frac{14}{3}\left[\begin{array}{c}{1} \\ {0} \\ {1} \\ {1}\end{array}\right]-\frac{5}{3}\left[\begin{array}{c}{0} \\ {-1} \\ {1} \\ {1}\end{array}\right]=\left[\begin{array}{c}{5} \\ {2} \\ {3} \\ {\frac{8}{3}}\end{array}\right]$(b) $\hat{x}=\left[\begin{array}{c}{\frac{1}{3}} \\ {\frac{14}{3}} \\ {-\frac{5}{3}}\end{array}\right]$

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 5

Least-Squares Problems

Vectors

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we want to find the orthogonal projection of the victor, be onto the span of the columns of a and then Ali Square solution for the system. A x equal. Be so we observe. First of all, that the columns are they are all for. You know, you could just simply check it. And so we can use this simple method. We ride the orthogonal projection of beings or behalf he's here and by the projection off, be onto the first column away, plus the projection off. Be onto the second column away, plus the projection off. Be onto the third column away. Now the computation of projection off be onto the columns is easy, so we get 1/3 the first column away last 14 14 over three. The second column of a minus five hours. If I thirst the third column of a putting this all together, we can just compute that their father protection behalf is given by five, two, three and six. So now, to find the least squares solution, we will need to solve this all the system a accept equal behalf, but because of the composition, we immediately see the disco efficient here. 1/3 4th in divided by three and minus 5/3 gives. Actually, the entries of exact so exact is given by 1/3 14 3rd and minus five develop battery.

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