Decompose Vector (2,3,-1): Find Distance to Subspace

[PhizKid]

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.

 

[LCKurtz]

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.

 

[vela]

Found the kernel of what?

 

[PhizKid]

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

 

[vela]

Your domain should be R³, not R².

 

[PhizKid]

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

 

[LCKurtz]

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 $(x,y,z)\rightarrow (x,2y,x)$.

[Edit]: There would be a 2 in the center.