Find Orthogonal Compliment to Span({[1 -1 1]T, [1 1 0]})

[03125]

Homework Statement

Find the orthogonal compliment to Span({[1 -1 1]^T, [1 1 0]})

Homework Equations

V(transpose)=Null(A)
u*v=<u,v>=U(transpose)v

The Attempt at a Solution

I need help understanding the notation of this problem, I am not sure what my MTX A will look like? I cannot find any problem like this in my book. This is a practice problem written by a different professor than the one teaching my class and his notation in general confuses me because I am not familiar with it.

I know that to solve for the transpose of A I reduce A to echelon form and then find the basis for the solution space Null(A) of Ax=0 Because V(transpose)=Null(A), which is the basis of the orthogonal complement of V (V being my row vectors v_1, v_2, ...,v_m of A)

Any help appreciated, thank you!

 

[lanedance]

how about using a cross product?

 

[03125]

how about using a cross product?

Perhaps you'd like to elaborate, that's not in my book and I don't know how that would work.

 

[lanedance]

Homework Statement

Find the orthogonal compliment to Span({[1 -1 1]^T, [1 1 0]})

also it should probably be
Find the orthogonal compliment to Span({[1 -1 1]^T, [1 1 0]^T})

each of the T's means transpose, as they represent column vectors

 

[03125]

also it should probably be
Find the orthogonal compliment to Span({[1 -1 1]^T, [1 1 0]^T})

each of the T's means transpose, as they represent column vectors

Yeah I wasn't sure how to make the ^T, thanks for showing me. As the problem is stated it is written "Find the orthogonal compliment to Span({[1 -1 1]^T, [1 1 0]})"

 

[lanedance]

lets call the 2 vectors in the span v1,v2. As we know the dimension of the space is 3 and there are 2 vectors, then a single vector will span the perpindicular complement, let's call it u.

you could approach this problem 2 ways

first is to use the fact that u is perpindicular to v1 & v2
<u,v1> = <u,v2> = 0
then write out the simultaneous equations and solve. This is in essence what you are doing with the matrix A.

the 2nd is to use the fact that the cross product (v1 x v2), gives a vector perpindicular to v1 & v2, which must be u (up to a multiplicative constant). Disregard this if you haven't covered cross products though.

 

[lanedance]

Yeah I wasn't sure how to make the ^T, thanks for showing me. As the problem is stated it is written "Find the orthogonal compliment to Span({[1 -1 1]^T, [1 1 0]})"

its probably a typo, they should both have T's i think