Project 5

For this project, you may want to watch the four subspaces video and the determinants/eigenvalues video.

1. For the matrix below, compute the row space, column space, nullspace and left nullspace using the theorem from class as done in the video. In other words compute these all using the appropriate span of vectors.

(1)
\begin{align} A=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6\\ 7 & 8 & 9 & 10 & 11 & 12\\ 9 & 8 & 7 & 6 & 5 & 4 \\ 3 & 2 & 1 & 0 & -1 & -2\\ 17 & 18 & 19 & 20 & 21 & 22 \end{pmatrix} \end{align}

2. Check your answers to the previous problem using Sage's built in functions to find the appropriate spaces.

3. Find a basis for the column space of the following matrix using the built in sage command and using the reduced row echelon form. Are the two bases the same?

(2)
\begin{align} B=\begin{pmatrix} 2 & 3 & 5 & 7 & 11\\ 13 & 17 & 19 & 23 & 29\\ 31 & 37 & 41 & 43 & 47\\ 53 & 59 & 61 & 67 & 71 \end{pmatrix} \end{align}

4. Come up with your own $3\times 5$ matrix with rank 2, and do the following:
A) Use the theorem from class and Sage to determine the four fundamental subspaces of your matrix.
B) Check that you are correct using Sage's built in functions to find the subspaces.
C) Find the dimensions of the four fundamental subspaces and check that Theorem 4.6 from Chapter 3.4 holds.

5. Create a $3\times 7$ matrix that has no 0 entries but has a column space of dimension 1.

6. Hint: The code below will generate the matrix given in the problem:

def CircMat(n,m):
    L=[]
    for i in range(n):
        L.append([])
        for j in range(n):
            if i==j:
                L[i].append(m)
            else:
                L[i].append(1)
    return matrix(QQ,L)

and

A=CircMat(10,9)
(3)
\begin{align} \left(\begin{array}{rrrrrrrrrr} 9 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 9 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 9 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 9 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 9 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 9 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 9 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 9 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 9 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 9 \end{array}\right) \end{align}

a. Find the determinant of this matrix. Is this matrix nonsingular?
b. Find the characteristic polynomial of this matrix. Use this polynomial to determine the eigenvalues.
c. Use Sage to find the eigenvalues and eigenvectors of this matrix.
d. Pick one eigenvector and demonstrated that it is in fact an eigenvector for the corresponding eigenvalue. (see the definition if necessary)

7. Find a matrix that has the 7 eigenvalues: 2, 3, 4, 5, 6, 7, 8

9. Optional: Use Sage to find the expanded formula for a generic $4\times 4$ matrix. How many terms are there in this formula. Do you have a guess as to how to generalize the determinant formula for $n\times n$ matrices.