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Student ID: | _______________________________ |

Please show all your work! Total points: (longer than an actual exam)

- Consider the linear transform on ℜ
^{3}corresponding to dilation in the x-axis by a factor of 2 followed by reflection about the plane y = z.- Construct the matrix representing this transformation.
- Find the determinant of this matrix.
*-2* - Find all eigenvalues and eigenvectors of this transform.
*Anything along the x-axis (t,0,0), with λ=2.*

Anything along the line y=z in the yz plane, (0,t,t), with λ=1.

- Construct the matrix representing this transformation.
- Consider the transform on ℜ
^{3}corresponding to orthogonal projection onto the plane 2x - y - 3z = 0.- Find the determinant of the matrix corresponding to this transform.
*0* - Find all eigenvalues and eigenvectors of this transform.
- Find a basis for the columnspace of the matrix corresponding to this transform. (There are multiple correct answers.)

- Find the determinant of the matrix corresponding to this transform.
- Consider the transform on ℜ
^{3}corresponding to reflection about the plane x + 3y - z + 1 = 0.- Is this transform linear? How can you tell?
*No: does not map 0 to 0. (It's that pesky "+1"!)* - Find all eigenvalues and eigenvectors of this transform.

- Is this transform linear? How can you tell?
- Consider the following 5x7 matrix:
(matrix goes here)
- Find the rank of the matrix.
- Find a basis for the rowspace of the matrix.
- Find a basis for the columnspace of the matrix.
- What dimension is the nullspace?
- Find a basis for the nullspace of the matrix.

- Find the rank of the matrix.
- Using either the Vandermonde or Newton methods, find the lowest-degree polynomial that interpolates through the points (0,4), (1,3), (2,-1), (3,5).
- Consider the set of vectors (r,t) → r cos(t) + r sin(t) for all r and t, using the standard operations: (r,t) + (s,u) → (r+s) cos(t+u) + (r+s) sin(t+u) and k(r,t) → kr cos(t) + ks sin(t). Is this a vector space?