Linear Algebra I: Homework 10

Due: Friday, November 10, 2017
  1. For vectors and , is an inner product.
    1. Find all unit vectors in with respect to this new inner product.

      See Homework 7.2.a; this is similar but with a different magnitude. Unit vectors are the ellipse determined by , that is, the set of vectors;

    2. Find two different orthonormal bases for with respect to this new inner product.

      Our solution for (a) says that two vectors determined by angles and have dot product,

      This means that two of our unit vectors are orthogonal if and form a right angle (why?). So to come up with orthonormal bases we just have to pick an angle, and then another angle which is orthogonal to it:

      For example, we get two different bases if we pick and .

  2. If and , find a decomposition

    where is parallel to and is orthogonal to .

    Gram-Schmidt says that

    Then,

  3. For two matrices we can define the inner product,

    Are the vectors,

    orthogonal? Explain why or why not.

    Yes, because .