Quiz 7 Ma 322 005 Name ______________

1. Suppose u, v and w are nonzero vectors which are pairwise orthogonal. Show that u, v and w are linearly independent.

Solution : Suppose s*u + t*v + r*w = O. Dot with u and get s*(u dot u) + t*(v dot u) + r*(w dot u) = O dot u = 0. But u dot u > 0 (since u is not O) and v dot u = w dot u = 0 (given). So s = 0. Similarly, dotting with v and w gets t = r = 0. Hence u, v and w are linearly independent.

2. Find the projection of onto the scalar multiples of .

Solution . Calculate the projection matrix, then multiply it by

3. Why is the null space of a matrix the orthogonal complement of its row space?

Solution: The definition of the null space says that each vector in the null space is orthogonal to each row in the matrix (hence all vectors in its row space). By a theorem, the dimension of the null space + the dimension of the row space = the number of columns of the matrix, so the null space is the complement of the row space.

4. Find a basis for the orthogonal complement of the row space of the matrix

Solution . Since the orthogonal complement of A is the null space of A, a basis would be the 3 columns of the null space matrix

> matrix(4,3,[-1,-1,-1,1,0,0,0,1,0,0,0,1]);

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