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I have a two part question:

  1. (a) Find the least squares approximation to the solution of the system of equations:

$2x + y = 1$

$x – y = 3$

$x +2 y = 2$

(b) Let S be the plane passing through the origin O and the points

$ A = (2, 1, 1$)

$B = (1, –1, 2)$.

Use your answer to (a) to find the foot of the perpendicular dropped from $P = (1, 3, 2)$ to $S$.

I already have the answer to a)

$x=1.332$ $y=-.333$

If I've done it correctly.

I don't understand how to approach b though, if someone could help me step by step to answer this, but not give me the answer, I would be appreciative.

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1 Answer 1

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Note that $A$ and $B$ are the columns of the matrix that you would get if you set up part (a) as a matrix equation, $Mv=c$. So the plane, $S$, is the column space of the matrix, $M$. Also, $P$ in (b) is $c$ in (a). Given any vector $w$, $Mw$ is in the column space of $M$, and in part (a) you're just finding $w=(x,y)$ such that $Mw$ is the closest point to $c$ in the column space of $M$. In (b), you're trying to find the closest point to $P$ in the space $S$. Do you see how they tie up?

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I don't understand what you're referring to by "column space", my professor hasn't used those terms. –  Unknown Oct 25 '12 at 3:00
    
The column space of a matrix is the vector space spanned by the columns of the matrix. –  Gerry Myerson Oct 25 '12 at 3:11
    
I'm slowly understanding but I'm having a hard time visualizing it. –  Unknown Oct 25 '12 at 4:11
    
Visualizing what? –  Gerry Myerson Oct 25 '12 at 5:48

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