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MATH 310 Homework: Orthogonal Complements, Projections, and Orthonormal Bases, Assignments of Linear Algebra

University of Illinois - ChicagoLinear Algebra

A university-level mathematics homework assignment focusing on orthogonal complements, finding orthonormal bases, orthogonal projections, and the pythagorean theorem in inner product spaces. It also covers the relationship between orthogonal matrices and their product.

Typology: Assignments

2011/2012

Uploaded on 05/18/2012

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MATH 310
Homework due 07/25/2011
1. Determine the orthogonal complement of the subspace of R3given by:
V={(x, y, z)T: 2xโˆ’y+z= 0}
and find an orthonormal basis for it.
2. Find the orthogonal projection of R3onto the kernel of the map T:
R3โ†’Rdefined by T((x, y, z)T) = xโˆ’y+z.
3. State and prove the Pythagorean theorem for inner product spaces.
4. Show that if Aand Bare orthogonal matrices, then AB is an orthog-
onal matrix.
5. Use the method of least squares in order to find the best approximation
to a solution for the system:
3x+y= 1
xโˆ’y= 2
x+ 3y=โˆ’1
6. Find an orthonormal basis for the subspace of R4defined by:
V={(x, y, z, w )T: 3xโˆ’y= 0, x +y+w= 0}
1

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MATH 310

Homework due 07/25/

  1. Determine the orthogonal complement of the subspace of R^3 given by:

V = {(x, y, z)T^ : 2x โˆ’ y + z = 0} and find an orthonormal basis for it.

  1. Find the orthogonal projection of R^3 onto the kernel of the map T : R^3 โ†’ R defined by T ((x, y, z)T^ ) = x โˆ’ y + z.
  2. State and prove the Pythagorean theorem for inner product spaces.
  3. Show that if A and B are orthogonal matrices, then AB is an orthog- onal matrix.
  4. Use the method of least squares in order to find the best approximation to a solution for the system: 3 x + y = 1 x โˆ’ y = 2 x + 3y = โˆ’ 1
  5. Find an orthonormal basis for the subspace of R^4 defined by:

V = {(x, y, z, w)T^ : 3x โˆ’ y = 0, x + y + w = 0}