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Inner product from ON basis
Task number: 4430
Assume that B={(1,1)T,(2,−1)T} is an orthonormal basis of the space R2 with respect to some non-standard inner product.
In other words, you do not have a formula for the inner product, but you do know that some vectors are orthogonal to each other.
With respect to this inner product, determine:
- the value of ⟨(1,4)T|(2,0)T⟩,
- the set of vectors orthogonal to L{(4,1)T}.
Solution 1
It is necessary to determine the coordinates of the given vectors with respect to the basis B. (12121−140)∼(12120−33−2)∼(121201−123)∼(1032301−123)We get [(1,4)T]B=(3,−1)T and [(2,0)T]B=(23,23)T
Now ⟨(1,4)T|(2,0)T⟩=[(2,0)T]TB[(1,4)T]B=(23,23)(3,−1)T=43.
Solution 2
Let us denote the matrix assembled from the basis B by the symbolB=(121−1)
Observe that for any vector u∈R2 it holds that [u]B=B−1u=(121−1)−1u=(132313−13)u.
Then the formula for the inner product can be derived as follows: ⟨u|v⟩=[v]TB[u]B=(B−1v)TB−1u=vT(B−1)TB−1u =vT(131323−13)(132313−13)u=vT(29191959)u=19(2u1v1+u1v2+u2v1+5u2v2)
Indeed: ⟨(1,4)T|(2,0)T⟩=19(2⋅1⋅2+1⋅0+4⋅2+5⋅4⋅0)=19(4+0+8+0)=43
Answer
The value of the inner product ⟨(1,4)T|(2,0)T⟩ is 43.Solution
Using the calculations from the previous section, every vector u orthogonal to (4,1)T must satisfy (4,1)(29191959)u=0
That is, (1,1)u=0, which (taken as a system of one equation with two unknowns) yields c(1,−1)T.
Answer
The vectors orthogonal to (4,1)T form the line c(1,−1)T.