Extra problems for
HW #26
 
  1. Let A be an n x n matrix.
    1. Prove v is in null(A) iff it is orthogonal to every row of A.
    2. Prove that v is orthogonal to every row of A iff it is orthogonal to row(A.).
    3. Conclude that null(A) is the orthogonal complement of row(A).
    4. Use Theorem 5.9 to conclude that row(A) is the othogonal complement of null(A).
  2. True or false: If W is a subspace of R^n, then every vector in R^n is either in W or the orthogonal complement of W. Prove your answer.
    1. Give an example of a 2-dimensional subspace W of R^3 other than the xy-, xz-, and yz-planes.
    2. Find vectors v and w such that w is in your subspace W from part (a), v is orthogonal to w, and v is neither in W nor in the orthogonal complement of W.
    1. Give an example of a 2-dimensional subspace W of R^4.
    2. Find vectors v and w such that w is in your subspace W from part (a), v is orthogonal to w, and v is neither in W nor in the orthogonal complement of W.