- #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?
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?