Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 1 down vote accepted

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?

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.