Finding a Basis for the Orthogonal Complement of W

[madglover]

Homework Statement

Let W be the subspace of R^3 spanned by the vectors
v1=[2,1,-2] and v2=[4,0,1]
Find a basis for the orthogonal complement of W

Homework Equations

None

The Attempt at a Solution

I can do this question except for the fact when i get the matrix in form

[2,1,-2/0
4,0,1/0]

i get to this bit but saying the complement is equal to (col(A))=null(A^T)

i can't perform reduced matrix stuff when there is not an odd number at the front i just can't do it can someone start of the next step for me? if so i can do the rest of the question

it is a resit i have tomorrow and we were told today that this question is in it, so it is worth a lot for me to know it by then

 

[madglover]

for example if someone could show me even how to reduce

[4,3
2,1]
i would be fine

i just don't understand how it is done

 

[madglover]

ooo i might have it i just read that i can interchange rows if so i am fine as i can let it become

1,-2,2/0
0,1,4/0

which i can easily reduce :D

grrr rows not columns

 

[Defennder]

You don't even have to perform row reduction. It is apparent by inspection that v1 and v2 are linearly independent (one is not a multiple of the other). So what you need to do is to find an orthogonal basis for W using a well known algorithm, and use a vector operation to find the vector orthogonal to both of them.

 

[madglover]

ok i am very confused now i have the basis

span{0,4,1}

is that right i managed to reduce it i think to
1 0 0/0
0 1 -4/0

 

[madglover]

You don't even have to perform row reduction. It is apparent by inspection that v1 and v2 are linearly independent (one is not a multiple of the other). So what you need to do is to find an orthogonal basis for W using a well known algorithm, and use a vector operation to find the vector orthogonal to both of them.

well now i am worse for wear, how would i do that? i thought the only way to find the complement to W is to use row reduction

 

[HallsofIvy]

The cross product of two vectors is perpendicular to both and so spans the subspace (in R³) orthogonal to the subspace spanned by the two vectors.

 

[Defennder]

Oh wait, you don't even need to find an orthogonal basis for W. The cross product would do. Silly me.