an orthogonal set is linearly........
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Answer:
I recently posted a question about this; now that I've acquired some new info, I have some follow up questions about extending a basis (I'm not too sure if this is actually the name for it, so my apologies if it isn't.)
Let's say I have a family ((1,1,1)) and wanted to add vectors such that the family is a basis of
R
R
3
3
. Tt's rather obvious that I could easily find a couple of vectors that would make the family linearly dependent, but I'm looking for a method that I could depend on.
I've seen a couple of ways of doing this, neither of which I understand fully.
The first is using dot products. Let's call the vectors I want to create a,b
∈
∈
R
R
. I want an a and a b such that a
⋅
⋅
(1,1,1) = 0 and that b
⋅
⋅
(1,1,1) = 0. If I've understood things correctly, this is because if the dot product of two vectors is 0, then they are orthogonal. If a and b are both orthogonal to (1,1,1), then ((1,1,1), a, b) is linearly independent, and therefore a basis on
R
R
3
3
. Is that all true?
If so, then I suppose I understand this solution conceptually, but I'm not sure how to go about it otherwise.
The second method I've found is by creating a matrix A out of my vectors, create a basis of the nullspace N(A), and then go from there. Honestly, I don't understand much of anything here. I know how to create a matrix from my 3 vectors (2 of which are unknown). I also understand that the nullspace of A is the set of vectors such that, if multiplied with A, the product is the zero vector (if this is also wrong, please correct me.) I also understand that if my matrix is built of linearly independent vectors, the nullspace is simply the zero vector. However, I don't understand what it means to create a basis of the nullspace, and thus don't know how to carry on.
The last thing that bothers me is, I don't see how these two methods are any more powerful/helpful than proving that having any linear combination of the vectors equaling zero implies that the scalars multiplying the vectors is also zero.
Any help is greatly appreciated, thank you.
Step-by-step explanation: