Linear Algebra: Span & Orthogonal Vectors

In summary, the method for orthogonalizing a set of vectors is to find two orthonormal vectors that are in the span of the given vectors.
  • #1
Xkaliber
59
0
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?
 
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  • #2
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.
 
  • #3
Dick said:
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?
 
  • #5
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.
 

1. What is the definition of a span in linear algebra?

The span of a set of vectors in linear algebra is the set of all possible linear combinations of those vectors. This means that the span includes all vectors that can be created by multiplying each vector by a constant and adding them together.

2. How do you know if two vectors are orthogonal?

Two vectors are considered orthogonal if their dot product is equal to zero. This means that the angle between the two vectors is 90 degrees and they are perpendicular to each other.

3. Can a set of orthogonal vectors span a vector space?

Yes, a set of orthogonal vectors can span a vector space. In fact, an orthonormal basis, which is a set of orthogonal vectors with a length of 1, is often used to span a vector space in linear algebra.

4. How can you find the orthogonal complement of a subspace?

The orthogonal complement of a subspace can be found by using the Gram-Schmidt process. This involves finding a set of orthogonal vectors that span the subspace and then normalizing them to create an orthonormal basis.

5. Why is the concept of span important in linear algebra?

The concept of span is important in linear algebra because it allows us to understand and manipulate vector spaces. It helps us to visualize how different vectors can combine to create new vectors and how these vectors can span a larger space. This is crucial in many applications, such as solving systems of linear equations and performing transformations in geometry.

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