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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. (12121140)(12120332)(121201123)(1032301123)

    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=(1211)

    Observe that for any vector uR2 it holds that [u]B=B1u=(1211)1u=(13231313)u.

    Then the formula for the inner product can be derived as follows: u|v=[v]TB[u]B=(B1v)TB1u=vT(B1)TB1u =vT(13132313)(13231313)u=vT(29191959)u=19(2u1v1+u1v2+u2v1+5u2v2)

    Indeed: (1,4)T|(2,0)T=19(212+10+42+540)=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.
Difficulty level: Moderate task
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