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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)).