This is a 2-hour test. You can use the book and the WebNotes but you cannot use the material not covered in class. If you want to refer to a result not in the Notes, you need to provide a complete proof of it otherwise your solution will be considered wrong.

1. Prove that if vector b is orthogonal to the column space of matrix A then v=0 is a least squares solution of the system Av=b.

2. Show that functions sin(x), xsin(x) are linearly independent in C[0,1] and find the projection of the function x2 onto the subspace spanned by these functions. What is the distance from x2 to this subspace?

3. Find a basis of the subspace in R4 spanned by the vectors (1,2,3,4), (2,3,4,5), (4,5,6,7), (2,2,2,2), (5,4,3,2).

4. Find the rank of the matrix

[ t 1 1 1 1 ]
[ t 1 1 1 t ]
[ 1 t t t 1 ]
[ t 1 1 t t ]

depending on t.