(answers in web view)

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Please show all your work! Total points: 40

1. Consider the linear transform on ℜ³ corresponding to contraction in the x-axis by a factor of 1/2 followed by rotation of 30°(=π/6) counter-clockwise about the z-axis.
   1. Find the determinant of the standard matrix for this transform. [2]
      1/2
   2. Construct the matrix representing this transformation. [4]
      

      √3/4	-1/2	0
      1/4	√3/2	0
      0	0	1

      (the order of matrix multiplication is important!)
2. Consider the class of rotations about the origin in ℜ², by an angle of θ. For every value of θ between 0 and 2π, find all eigenvalues and eigenvectors of the rotation. This question may be solved geometrically, without explicitly constructing the transform matrices. [8]
   Geometric reasoning is probably a better approach here than algebraic manipulation using the characteristic equation (det(λI-A)=0):
   When θ = 0 or 2π, all x in ℜ² are eigenvectors, with λ = 1.
   When θ = π, all x in ℜ² are eigenvectors, with λ = -1.
   For all other θ, there are no eigenvectors.
3. Consider the following 4x6 matrix A:

   1	-2	2	-1	0	2
   -3	6	-1	0	2	2
   2	-4	1	0	2	0
   1	2	3	0	2	-2

   1. Find the rank of the matrix. [4]
      4
   2. Find a basis for the rowspace of the matrix. [2]
      {R₁, R₂, R₃, R₄} (i.e., all the rows; the matrix is full-rank.)
   3. Find a basis for the columnspace of the matrix. [2]
      {C₁, C₂, C₃, C₄}
   4. What dimension is the nullspace? [1]
      2
   5. Find all vectors x for which Ax = 0. [2]
      {C₅, C₆} form a basis for the nullspace, so all linear combinations of those two columns.
4. Using either the Vandermonde or Newton methods, find the lowest-degree polynomial that interpolates through the points (-1, 2), (0, -1), (1, 0), (2, 1). Check your answer. [6]
   -1 -(1/3) x +2 x² -(2/3) x³
5. Consider the unit circle in ℜ², i.e., all vectors with unit length.
   1. Using the standard operations on ℜ², is the unit circle a vector space? (If so, find the zero element (additive identity) and negatives (additive inverses). If not, it's sufficient to demonstrate one axiom that is false.) [4]
      No: (0,0) is not in the set.
   2. Let's redefine vector addition on the unit circle as
      (cos(θ₁), sin(θ₁)) + (cos(θ₂), sin(θ₂)) = (cos(θ₁ + θ₂), sin(θ₁ + θ₂)).
      Also, redefine scalar multiplication on the unit circle as
      k(cos(θ), sin(θ)) = (cos(kθ), sin(kθ)).
      With these new operations, is the unit circle in ℜ² a vector space? [5]
      Yes:
      e.g., zero element is (cos(0),sin(0)) = (1,0).
      Negative is -(cos(θ), sin(θ)) = (cos(2π-θ), sin(2π-θ)).