Problem 1 [10 pts.]
(a) 
	[1 pt.] Correct answer: true
	[2 pts.] Justification
		Okay if case of zero ideal is not mentioned
		Mention that any nonzero polynomial in I with minimal degree is a generator
		Use the division algorithm to justify the above claim
(b) 
	[1 pt.] Correct answer: false
	[1 pt.] Justification
(c)
	[1 pt.] Correct answer: true
	[2 pts.] Justification
		k[x]/(q(x)) is a field 
		k[x]/(q(x)) has dim 5 as vector space over k
(d)
	[1 pt.] Correct answer: true
	[1 pt.] Justification

Problem 2 [3 pts.]
	[1 pt.] Correct answer: false
	[1 pt.] Correct setup: k,L,W,subset chosen appropriately
	[1 pt.] Correct justification

Problem 3 [7 pts.]
Let S={v_1,...,v_n} and W the subspace of V spanned by S
	[1 pt.] Stating (c) = (a) + (b)
	[3 pts.] (a) --> (b)
	[3 pts.] (b) --> (a)

Problem 4 [6 pts.]
Let S be a subset of V with m>n elements
Method 1: 	
	Assume for the sake of contradiction that S is linearly independent
	Use Problem 3 to argue that W:=Span(S) has dim m
	Use Homework 1 Problem 2 to show W \iso k^r for some r \leq n
	Derive a contradiction using Homework 1 Problem 4(a) (uniqueness of dim)
Method 2:
	Choose S' a maximal linearly independent subset of S
	Use Problem 5 to argue that S' has size at most n
	Deduce that S is linearly dependent
Method 3: 
	Show that any linearly independent subset of V can be extended to a basis

Problem 5 [12 pts.]
(a)
	[1 pt.] Mention that {v,T(v),...,T^n(v)} has n+1 elements
	[1 pt.] Find a linear dependence relation
(b)
	[1 pt.] Show that p(T) commutes with vector addition
	[1 pt.] Show that p(T) commutes with scalar multiplication
(c)
	[1 pt.] Correct use of convolution
	[1 pt.] Show work
(d)
	[1 pt.] Pick a basis {v_1,...,v_n} of V
	[1 pt.] Use (a) for each i to find a nonzero p_i(x) in k[x] such that p_i(T)(v_i)=0
	[1 pt.] Invoke p(x):=p_1(x)...p_n(x)
	[2 pts.] Show that p(x) satisfies p(T)=0
		Use that p_i(T)(v_i)=0
		Note that p_i(T) and p_j(T) commute
	[1 pt.] Explain why p(x) is nonzero in k[x]

Problem 6
	[2 pts.] Use Problem 5(d) to choose p(x) in C[x] nonzero (monic) of minimal degree such that p(T)=0
	[2 pts.] Use fundamental theorem of algebra to write p(x)=(x-\lambda)q(x) for some \lambda in C
	[2 pts.] Argue that q(T) \neq 0 for degree reasons and so q(T)(v) \neq 0 for some v in V
	[2 pts.] Show that \lambda is eigenvalue of T with eigenvector q(T)(v)

Problem 7 [3 pts.]
	Write f(x) in terms of coefficients
	Find the images of 1,...,x^{d-2} mod f(x)
	Find the image of x^{d-1} mod f(x)
	Write everything in matrix form

Problem 8 [8 pts.]

Problem 9 [8 pts.]
Let A be the matrix 
0 -1
1 0
	Show that A is diagonalizable over C
		[2 pts.] Show that A has eigenvalues i and -i
		[2 pts.] Show that A has two linearly independent eigenvectors
		[2 pts.] Relate diagonalizability to existence of an eigenbasis
	Show that A is not diagonalizable over R
		[1 pt.] Mention that a 2x2 matrix must have all real eigenvalues to be diagonalizable over R
		[1 pt.] Explain why the above is true