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Please show all your work! Total points: 40

- Consider the linear transform on ℜ
^{3}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.- Find the determinant of the standard matrix for this transform.
__[2]__*1/2* - Construct the matrix representing this transformation.
__[4]__√3/4 -1/2 0 1/4 √3/2 0 0 0 1

- Find the determinant of the standard matrix for this transform.
- Consider the class of rotations about the origin in
ℜ
^{2}, 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 ℜ^{2}are eigenvectors, with λ = 1.

When θ = π, all x in ℜ^{2}are eigenvectors, with λ = -1.

For all other θ, there are no eigenvectors. - 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 - Find the rank of the matrix.
__[4]__*4* - Find a basis for the rowspace of the matrix.
__[2]__*{R*_{1}, R_{2}, R_{3}, R_{4}} (i.e., all the rows; the matrix is full-rank.) - Find a basis for the columnspace of the matrix.
__[2]__*{C*_{1}, C_{2}, C_{3}, C_{4}} - What dimension is the nullspace?
__[1]__*2* - Find all vectors x for which Ax = 0.
__[2]__*{C*_{5}, C_{6}} form a basis for the nullspace, so all linear combinations of those two columns.

- Find the rank of the matrix.
- 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 x*^{2}-(2/3) x^{3} - Consider the unit circle in ℜ
^{2}, i.e., all vectors with unit length.- Using the standard operations on ℜ
^{2}, 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.* - Let's redefine vector addition on the unit circle as

(cos(θ_{1}), sin(θ_{1})) + (cos(θ_{2}), sin(θ_{2})) = (cos(θ_{1}+ θ_{2}), sin(θ_{1}+ θ_{2})).

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 ℜ^{2}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)).

- Using the standard operations on ℜ