For this week's project please watch the 6th video about the Four Fundamental Subspaces and then complete the following questions.

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

(1)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)4. Come up with your own $5\times 5$ matrix, and do the following:

A) Use the theorem in the video 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 rank of your matrix and then find the dimensions of all of the fundamental subspaces.

5. Come up with your own singular $4\times 4$ matrix. Find the dimensions of the four fundamental subspaces using any method you want.

6. Come up with your own $3\times 5$ matrix. Find the dimensions of the four fundamental subspaces using any method you want.

7. Do you have any guesses about relationships between the dimensions of the subspaces and the matrix?

8. Can you figure out which matrices have nullspace $N(A)=\{\mathbf{0}\}$?

9. Create a $2\times 7$ matrix with a column space of dimension 1.