(answers in web view)

Name:	_______________________________
Student ID:	_______________________________

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

1. Consider the linear transform on ℜ³ corresponding to dilation in the x-axis by a factor of 2 followed by reflection about the plane y = z.
   1. Construct the matrix representing this transformation.
      
   2. Find the determinant of this matrix.
      -2
   3. 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.
2. Consider the transform on ℜ³ corresponding to orthogonal projection onto the plane 2x - y - 3z = 0.
   1. Find the determinant of the matrix corresponding to this transform.
      0
   2. Find all eigenvalues and eigenvectors of this transform.
      
   3. Find a basis for the columnspace of the matrix corresponding to this transform. (There are multiple correct answers.)
3. Consider the transform on ℜ³ corresponding to reflection about the plane x + 3y - z + 1 = 0.
   1. Is this transform linear? How can you tell?
      No: does not map 0 to 0. (It's that pesky "+1"!)
   2. Find all eigenvalues and eigenvectors of this transform.
4. Consider the following 5x7 matrix: (matrix goes here)
   1. Find the rank of the matrix.
      
   2. Find a basis for the rowspace of the matrix.
   3. Find a basis for the columnspace of the matrix.
   4. What dimension is the nullspace?
   5. Find a basis for the nullspace of the matrix.
5. 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).
6. 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?