# Decomposing a vector

## Homework Statement

There is a subspace that contains all the vectors in the form (x, 2y, x). Decompose the vector (2, 3, -1) into a sum of an element from the orthogonal complement of this subspace and an element from the subspace. Find the distance from (2, 3, -1) to this subspace.

## The Attempt at a Solution

To find the orthogonal complement of this subspace, I found the kernel, which in this case happens to only contain the zero vector. That means only a particular solution exists, but obviously (2, 3, -1) is not a particular solution, so I'm not sure how to decompose this, much less find the distance.

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LCKurtz
Homework Helper
Gold Member

## Homework Statement

There is a subspace that contains all the vectors in the form (x, 2y, x). Decompose the vector (2, 3, -1) into a sum of an element from the orthogonal complement of this subspace and an element from the subspace. Find the distance from (2, 3, -1) to this subspace.

## The Attempt at a Solution

To find the orthogonal complement of this subspace, I found the kernel, which in this case happens to only contain the zero vector. That means only a particular solution exists, but obviously (2, 3, -1) is not a particular solution, so I'm not sure how to decompose this, much less find the distance.
Why do you say the kernel is only the zero vector? ##(x,2y,x)=x(1,0,1)+y(0,2,0)## is clearly only two dimensional.

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vela
Staff Emeritus
Homework Helper
Found the kernel of what?

The kernel of
##\ \left( \begin{array}{ccc}
1 & 0 \\
0 & 1 \\
1 & 0 \end{array} \right)##

vela
Staff Emeritus
Homework Helper
Your domain should be R3, not R2.

So then it's:
##\ \left( \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
1 & 0 & 0 \end{array} \right)##
?

LCKurtz
Homework Helper
Gold Member
So then it's:
##\ \left( \begin{array}{ccc}
1 & 0 & 0 \\
0 & \color{red}{2} & 0 \\
1 & 0 & 0 \end{array} \right)##
?
Yes, that matrix represents the transformation ##(x,y,z)\rightarrow (x,2y,x)##.

: There would be a 2 in the center.

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