Linear Algebra: Span & Orthogonal Vectors

[Xkaliber]

Hi all,

I am having a little trouble understanding the concept of span. I do realize the definition of the span of two vectors is all possible linear combinations of the two vectors but I am trying to make the concept understandable with regards to the problems I have been assigned.

Before I give the problem, let me first ask a question. Since I do understand the concept of column space, can I think of a span of two vectors in a similar manner. For example,
is the column space of
[1 1
1 0
2 1]
equal to the span of the column vectors v1 = [1 1 2] and v2 = [1 0 1] ?Anyway, here is the problem along with my solution.

Let column vectors v1 = [1 1 2] and v2 = [1 0 1]
Find mutually orthogonal vectors u1 and u2 such that the span of {v1, v2} is the same as the span of {u1, u2}.

First, I check to see if the vectors are multiples of each other. Since they are not, I know that the two vectors make a plane, which is the span of the vectors. I use the cross product to find the equation of the plane which is i + j - k = 0 Since I must find two orthogonal vectors that are in the span of the two given vectors, I am basically looking for two orthogonal vectors that satisfy the above equation of the plane. So if I take v1 = u1, then I can choose u2 = [1 -1 0]

Is this correct?

 

[Dick]

I think you understand the concept very well. There are other ways to orthogonalize a set of vectors that don't involve the cross product but that will work.

 

[Xkaliber]

I think you understand the concept very well. There are other ways to orthogonalize a set of vectors that don't involve the cross product but that will work.

Could you please elaborate on the method at which you hinted? Does it work in dimensions greater than 3?

 

[ZioX]

http://en.wikipedia.org/wiki/Gram_schmidt

This is the method for determining an orthogonal/orthonormal basis given a spanning set.

 

[Dick]

It's called Gram-Schmidt. And, yes it works in dimensions not equal to 3, where you don't have a cross product. You basically use the dot product to remove parallel components of vectors. I'll let you look it up.