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Orthogonality

Part 3: Orthonormal bases

Again we let W denote the column space of the matrix A from Part 1. Until you get to step 5, the vector y is the same vector that you used in Part 2.

  1. Construct U as a matrix whose columns are proportional to the columns of A and each has length 1. Explain why the columns of U are an orthonormal set of vectors.

  2. Calculate UTU and UUT. How do they differ? What does this calculation have to do with the preceding step?

  3. Explain why projWy = UUTy. Use this formula to find p = projWy. Compare your result to the one calculated in Part 2.

  4. Define z to be y - p. Explain how you know that z is in Wperp. Confirm that this is the case.

  5. Redefine y as a new random vector in R8. Find the vector in W that is closest to y, and find the distance from y to W.

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