This rubric assigns credit based on the effort required to do things. Please give me any feedback you might have.

Problem 1 [18 pts.]
(a)
Say the matrix for T with respect to the standard bases of k^m and k^n is A
	[2 pts.] Use row and column operations to put A in Smith normal form (SNF) -- block form with id matrix in the upper left and 0's elsewhere
	[1 pt.] Mention the case of the zero matrix
	[1 pt.] Count the # of vanishing columns -- let it be r
	[2 pts.] Explain why dim_k ker(T) \leq r (span)
	[2 pts.] Explain why dim_k ker(T) \geq r (linear independence)
(b)
	[2 pts.] Explain why dim_k im(T) \leq m-r
	[2 pts.] Explain why dim_k im(T) \geq m-r
	[1 pt.] Compare dimensions of k^n/im(T) and k^{n-m+r}
(ci)
	[1 pt.] Explain why r=0
	[1 pt.] Relate m and n
(cii)
	[1 pt.] Explain why n-m+r=0
	[1 pt.] Relate m and n
(ciii)
	[1 pt.] Combine (ci) and (cii)

Problem 2 [12 pts.]
Let H be a k-subspace of k^n
	Base step
		[1 pt.] Appeal to the fact that k is a field
		[3 pts.] Explain why k being a field matters
			Assume H is nonzero, so that it contains a nonzero element
			H contains 1
			H is all of k
	Inductive hypothesis
		[1 pt.] Give a clear statement
	Inductive step
		[1 pt.] Invoke coordinate projection map \pi: k^{n+1} --> k
		[1 pt.] Identify that ker(\pi) is isomorphic to k^n
		[2 pts.] Apply inductive hypothesis to H \cap ker(\pi) to get isomorphism with some k^m for m \leq n
		[3 pts.] Argue that H is isomorphic to k^m or k^{m+1}

Problem 3 [10 pts.]
[3 pts.] for each of the following, up to 9 points
	(a) --> (b)
	(a) --> (c)
	(b) --> (a)
	(b) --> (c)
	(c) --> (a)
	(c) --> (b)
[1 pt.] Logical equivalence completely justified (can be done with minimum of three implications)


Problem 4 [8 pts.]
Let V be a finitely generated k-vector space
(a)
	[1 pt.] Use the definition of finitely generated to write V as the span of an m-element subset
	[1 pt.] Appeal to Problem 3(c) to write V as a quotient -- say k^m/Z
	[1 pt.] Appeal to Problem 2 to get an isomorphism Z \iso k^r for some r \leq m
	[1 pt.] Appeal to Problem 1(b) to get an isomorphism k^m/Z \iso k^{m-r} -- hence V \iso k^{m-r} and m-r is the desired value of n
		Technical remark: Problem 1(b) deals with images of linear maps, but every linear subspace is the image of its (linear) inclusion map
	Explain why n is unique
		[1 pt.] Reduce to the case k^s \iso k^t
		[1 pt.] Appeal to Problem (ciii) to conclude s=t
	[1 pt.] total allowed for using the coordinate vector approach
(b)
Let W be a k-subspace of V
Method 1:
	[1 pt.] Appeal to Problems 4(a)	and 2 to conclude W \iso k^m for some m
	[1 pt.] Explain why W is then finitely generated

Method 2:
	[1 pt.] Examine iterated spans of elements in W
	[1 pt.] Argue that the iteration process must stop at some finite step since V has finite dimension (this is Problem 4 on Homework 2)

Problem 5 [5 pts.]
Let T: V --> W be a k-linear map
	[1 pt.] Appeal to Problem 4(a) to conclude that V \iso k^n for some n
	[1 pt.] Appeal to Problem 4(b) to conclude that ker(T) \iso k^m for some m \leq n
	[2 pts.] Use factoring triangle to conclude that im(T) \iso k^{n-m}
	[1 pt.] Compare dimensions to get the result

There is an alternative method that extends a basis of ker(T), but you need to explain how to do this using the factoring triangle for the quotient
Yet another method is to appeal directly to Problem 1, though beware that many people basically used this result to do Problem 1
Yet another method is to appeal to the finite dimensionality of im(T) (as a subspace of W) and then look at some preimages of a basis for im(T)

Problem 6 [10 pts.]
Method 1:
Let ev_0,ev_1,ev_2: P_6(k) --> k be the k-linear maps that "plug in" the appropriate value
Let V be the k-subspace of P_6(k) consisting of those polynomials that vanish at 0, 1, and 2
	[1 pt.] Identify V as ker(T) for T = (ev_0,ev_1,ev_2): P_6(k) --> k^3 (note that don't need to worry about ev_2 if 2=0)
	Case 1: 2 \neq 0 in k
		[1 pt.] Write the matrix A for T
		[1 pt.] Reduce A
		[2 pts.] Find a basis for V=ker(T) and write general element of V
		[2 pts.] Identify elements of degree 6
	Case 2: 2=0 in k
		[1 pt.] Write the matrix A for T and reduce A
		[1 pt.] Find a basis for V=ker(T) and write general element of V
		[1 pt.] Identify elements of degree 6
	[1 pt.] may be lost overall for failing to make sure that linear combination has degree 6 (and not some smaller degree)

Method 2:
Let S be the set of degree-6 polynomials over k that vanish at 0, 1, and 2
	[1 pt.] State that vanishing at c is equivalent to divisibility by c
	[2 pts.] Justify that vanishing at c is equivalent to divisibility by c
	[1 pt.] State that x and x-1 are relatively prime in k[x]
	[2 pts.] Justify that x and x-1 are relatively prime in k[x]
	Case 1: 2 \neq 0 in k
		[1 pt.] State and justify that x and x-2 are relatively prime in k[x]
		[1 pt.] Identify S as the polynomials in P_6(k) divisible by x(x-1)(x-2)
	Case 2: 2=0 in k
		[1 pt.] State and justify that x and x-2 are not relatively prime in k[x]
		[1 pt.] Identify S as the polynomials in P_6(k) divisible by x(x-1) (and not by x^2(x-1))

Problem 7 [10 pts.]
Let m be the L-dimension of V
Method 1:
	[1 pt.] Choose an L-basis v_1,...,v_m of V
	[1 pt.] Choose a k-basis x_1,...,x_n of L
	[3 pts.] Explain why the k-span of the set S={x_1v_1,...,x_nv_1,...,x_nv_m} is V
		Use spanning condition for V
		Use spanning condition for L
		Do some algebra
	[3 pts.] Explain why S is k-linearly independent
		Use linear independence condition for V
		Use linear independence condition for L
		Do some algebra
	[2 pts.] Count dimensions to get the result

Method 2:
	[1 pt.] Choose an isomorphism f: V \iso L^m of L-vector spaces
	[1 pt.] Choose an isomorphism g: L \iso k^n of k-vector spaces
	[6 pts.] Explain why we get an isomorphism V \iso (k^n)^m \iso k^{nm} of k-vector spaces
		Put a k-vector space structure on V
		Upgrade f to a k-linear map
		Argue that f is a k-linear isomorphism
		Explain why (k^n)^m \iso k^{nm} as k-vector spaces
	[2 pts.] Count dimensions to get the result

Problem 8 [13 pts.]
(a)
	Explain why k is a vector space over F_p for some prime p -- p is called a characteristic for k
		[3 pts.] Argue that the minimal positive multiple of 1_k that vanishes is a prime p
			Let n be the minimal positive multiple of 1_k that vanishes
			n can't be 1 since 0_k and 1_k are distinct
			n must be prime since if not we could factor it
		[1 pt.] Explain why this gives an F_p-vector space structure on k
	[1 pt.] Explain why k has finite dimension over F_p
	[3 pts.] Explain why |k|=p^r for some r \geq 1
		Let r be the dimension of k over F_p (can't be 0 since k has at least two elements)
		State that there is an isomorphism between k and F_p^r
		Measure the size of both sides
(b)
	[3 pts.] Explain why p=q
		Method 1: argue that k has a unique characteristic by supposing for contradiction that p \neq q
			Using that gcd(p,q)=1, choose integers a,b such that ap+bq=1 in Z
			Show that 0_k = 1_k by doing some algebra
			Appeal to the field axiom that says 0_k and 1_k must be distinct
		Method 2: use divisibility properties of primes to show that p and q both divide each other hence are equal
	[2 pts.] Explain why r|s
		Appeal to Problem 7 to write dim_{F_p} L = rm for m = dim_k L
		Appeal to uniqueness of dimension over F_p to conclude s = rm