- #1
ichabodgrant
- 49
- 0
Let u = [1, 2, 3, -1, 2]T, v = [2, 4, 7, 2, -1]T in ℝ5.
Find a basis of a space W such that w ⊥ u and w ⊥ v for all w ∈ W.
I think the question is quite easy. Given this vector w in the space W is orthogonal to both u and v. I can only think of w being a zero vector. But would this be too trivial?
wTu = wTv = 0
wT(u - v) = 0
I believe there is something wrong. Things above are what I can compute...
Is there any other way to solve this question?
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Oh, I've thought of another way.
Set a augmented matrix
[1 2 3 -1 2 | 0]
[2 4 7 2 -1 | 0]
~
[1 2 3 -1 2 | 0]
[0 2 4 3 -3 | 0]
Let x5 = t, x4 = s, x3 = z where t, s, z ∈ ℝ.
∴ x2 = 1.5t - 1.5s - 2z
x1 = -5t + 4s + z
Therefore, the solution set is {t[-5 1.5 2 0 0 1]T + s[4 -1.5 0 1 0]T + z[1 -2 1 0 0]T}.
The basis is {[-5 1.5 2 0 0 1]T, [4 -1.5 0 1 0]T, [1 -2 1 0 0]T}.Is this correct?
Find a basis of a space W such that w ⊥ u and w ⊥ v for all w ∈ W.
I think the question is quite easy. Given this vector w in the space W is orthogonal to both u and v. I can only think of w being a zero vector. But would this be too trivial?
wTu = wTv = 0
wT(u - v) = 0
I believe there is something wrong. Things above are what I can compute...
Is there any other way to solve this question?
-----------------------------------------------------------------------------------------------------
Oh, I've thought of another way.
Set a augmented matrix
[1 2 3 -1 2 | 0]
[2 4 7 2 -1 | 0]
~
[1 2 3 -1 2 | 0]
[0 2 4 3 -3 | 0]
Let x5 = t, x4 = s, x3 = z where t, s, z ∈ ℝ.
∴ x2 = 1.5t - 1.5s - 2z
x1 = -5t + 4s + z
Therefore, the solution set is {t[-5 1.5 2 0 0 1]T + s[4 -1.5 0 1 0]T + z[1 -2 1 0 0]T}.
The basis is {[-5 1.5 2 0 0 1]T, [4 -1.5 0 1 0]T, [1 -2 1 0 0]T}.Is this correct?
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