# Least Squares Approximation and Foot of the Perpendicular

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

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