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.