There are 8 questions, each is worth 2.5 points. I will grade rather "strict" this time, since all the problems are similar to the homework problems, and since everybody who wants to pass the class should be able to do all of the problems without any mistakes. Check your current amount of points here, and see the Syllabus for the table on how to convert points into a final grade (there will not be any exemption from the rule given in this table).

1. Write a certain system of equations as Ax=b, find the LDU decomposition of A, find c with Lc=b, and finally find x with DUx=c.

2. Determine whether a certain set is a subspace, and give the dimension and basis if it is a subspace.

3. Find the echelon form of a given matrix, the basic variables, the free variables, the solution to Ax=0, and finally the solution to Ax=b for a given b.

4. Decide whether some given vectors are linearly independent, orthogonal, and find the angles between them.

5. Find a basis and the dimension of each of the four fundamental subspaces of a given matrix.

6. Find the least-square solution to a given equation Ax=b as well as the projection of b onto the image of A.

7. Find all eigenvalues, eigenvectors, the trace, and the determinant of a certain matrix.

8. Proof a little statement concerning the material from the last week.