Linear Systems - Mathematics 214 - Fall 2007

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. 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), and v is orthogonal to w, and v is not in the orthogonal complement of W.
3. 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), and v is orthogonal to w, and v is not in the orthogonal complement of W.