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Orthogonality

Part 2: Orthogonal projections

In this part we let W denote the column space of the matrix A from Part 1.

  1. Explain why W is a 4-dimensional subspace of R8. What is a basis for W?

  2. Let y be a random vector in R8. Find the projection of y in the direction of A1, the first column of A.

  3. Find the projection projWy of y in the subspace W. (Hint: You can recycle the preceding step to find the projection of y in each of the column directions.)

  4. Write y as the sum of two vectors, one of which is in W, and the other of which is in Wperp. Explain how you know that there is only one way to do this.

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