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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} .$$$A=\left[\begin{array}{rr}{1} & {5} \\ {3} & {1} \\ {-2} & {4}\end{array}\right], \mathbf{b}=\left[\begin{array}{r}{4} \\ {-2} \\ {-3}\end{array}\right]$$

(a) $\hat{b}=\frac{4}{14} a_{1}+\frac{6}{42} a_{2}=\frac{2}{7}\left[\begin{array}{c}{1} \\ {3} \\ {-2}\end{array}\right]+\frac{1}{7}\left[\begin{array}{l}{5} \\ {1} \\ {4}\end{array}\right]=\left[\begin{array}{l}{1} \\ {1} \\ {0}\end{array}\right]$(b) $\hat{x}=\left[\begin{array}{l}{2} \\ {\frac{1}{7}}\end{array}\right]$

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 5

Least-Squares Problems

Vectors

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we observed that the columns of a roo for you know, therefore, we can easily compute their fucking a projection on be on the span of the columns of a So they're throwing a projection of me. The hat is just the projection on tow. The first column of a off me bluff. The projection onto the second column of a off B and this is just an easy computation. We get 95 by three. The first column, a one bluff Delacroix efficient is 12 the water by 24 times the second column away and putting all this together, we see that be hat is four minus 14 Now, to find the least square solution, we would need to solve the system a X hat, equal be hat. But we see immediately from the from the formula here what the interest of acts at our because these proficient here 9 91 every three and these other condition tear 12 year, October 24 are dentists of Accent. So immediately we see that exact is given by well $90.33 and 12 divided by 24 which is 1/2 and at the least, choir solution

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