## Math 24 Homework Assignments

Suggested but not for handing in

3.4 #5, 7, 10, 14
For 7 see example 3 and its discussion on page 192; for 10 see example 4.

4.4 #5, 6

5.1 #2, 3, 4, 10, 11, 14, 15, 17, 19, 22
For #14 use the relationship between det(A) and det(At).

5.2 #2, 3, 8, 9, 10, 13

Due Friday, May 26

3.2 #3, 4, 6, 8, 14

3.3 #2(a, b), 3(a, b), 4, 6, 7(a, b)
For #6 note that by T-1(1, 11) they mean the set of all elements T maps to (1, 11) (since T is not 1-1 there can be no proper T-1).

Due Friday, May 19

Section 2.5 #4, 5, 9, 12, 13

3.1 #2, 3, 4, 6

Proof Assignment 6

Exercise 8 from 2.5. That is, prove the following generalization of Theorem 2.3 (you'll want to look at the proof of 2.3 to do it):

Theorem. Let T:V→W be a linear transformation between finite-dimensional vector spaces V, W. Let β and β' be ordered bases for V, and let γ and γ' be ordered bases for W. Then [T]β'γ' = P-1[T]βγQ, where P is the matrix that changes γ' coordinates into γ coordinates, and Q is the matrix that changes β' coordinates into β coordinates.

Due Friday, May 12

Section 2.4 #2, 3, 4, 5, 6, 10, 13, 16, 20
Note that the definition of equivalence relation is on p. 551

Proof Assignment 5

Proposition. Let V, W be finite-dimensional vector spaces and let T:V->W be an isomorphism. Suppose that V' is a subspace of V. Then T(V') is a subspace of W and has the same dimension as V'.

Due Friday, May 5

Section 2.2 #5, 8, 10, 11, 13, 15

2.3 #4, 11, 12, 15, 17

No proof this week.

Due Friday, April 28

Section 2.1 #4, 5, 9, 10, 14, 15, 16, 17, 20, 26, 28, 29

Proof Assignment 4

Proposition. Let V be a finite-dimensional vector space, and T:V->V be linear. Then the following are equivalent:
(a) V = R(T) + N(T)
(b) V = R(T) [direct sum] N(T)
(c) R(T) [intersect] N(T) = {0}.

Hints: Be sure to explicitly state where finite-dimensionality is used. May want to structure this as (a) <-> (b) and (c) <-> (b), where (b) implies (a) and (c) by definition of direct sum (which along with + is defined on page 22).
More hints: the only ingredients you need are the definition of direct sum, what equality between dimension of a vector space and dimension of a subspace indicates about the space and subspace, the Dimension Theorem (page 70 in your book) and Proof Assignment 3 with N(T) and R(T) playing the roles of W1 and W2. Finite-dimensionality is a prerequisite to using many of those pieces.

Due Friday, April 21

Section 1.6 #2(a,b), 3(a,b), 5, 7, 11, 12, 14, 15, 22

Proof assignment 3

Prove the statement of Section 1.6 #29(a) and then 29(b) as a corollary. In other words, prove:

Proposition. If W1 and W2 are finite-dimensional subspaces of a vector space V, then the subspace W1+W2 is finite-dimensional and its dimension is dim(W1) + dim(W2) - dim(W1 intersect W2).
Corollary. If W1 and W2 are finite-dimensional subspaces of a vector space V such that V = W1+W2, then V is the direct sum of W1 and W2 if and only if dim(V) = dim(W1) + dim(W2).

You may assume the reader is familiar with the definition of sum and direct sum for vector spaces (though it is good to say things like "to show V is the direct sum it remains to prove ...") and with the result from Section 1.3 (p. 22, exercise 23) that the sum of two subspaces is a subspace (so for the proposition, you need to prove the dimension calculation only).

Due Friday, April 14

Section 1.2 #2, 3, 7, 8, 9, 13, 14, 15, 17
For #7, see example 3. For #9 and other proofs in homework (that is, proofs not singled out as "the proof assignment") it need not be written up in a fully formal way.

Section 1.3 #2, 4, 5, 8, 9, 14, 20, 23, 25

Section 1.4 #2(a,b,f), 3(a,b,c), 4(a,b,c), 6, 10, 11, 13, 15
For 6, you are working with a general F whose multiplicative identity is named 1 and additive identity named 0. For 13, prove the first sentence as a proposition and the second as its corollary (but again, not in a fully formal manner).

Section 1.5 #2(a,c,e), 3, 5, 7, 9, 11, 16

Proof Assignment 2: prove the following and write up.

Claim. Let W_1 and W_2 be subspaces of a vector space V. Then W_1 union W_2 is a subspace of V if and only if W_1 is a subset of W_2 or vice-versa.

Hint: do the "if and only if" in two separate directions. One will be immediate. For the other, try contradiction.

Due Friday, April 7 in pdf.

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