Math 3135: Spring 2017

Graded assignments:

• Vector spaces [tex] [submit on D2L as "Vector spaces homework"]
• Basis [tex] [submit on D2L as "Basis homework"]
• Linear transformations [tex] [submit on D2L as "Linear transformations homework"]
• Kernel and image [tex] [submit on D2L as "Kernel and image homework"]
• Dual vector space [tex] [submit on D2L as "Dual homework"]
• Determinants homework [tex] [submit on D2L as "Determinant homework"]
• Eigenvectors and eigenvalues homework [tex] [submit on D2L as "Eigenvalues and eigenvectors homework] — This assignment is optional: it will not affect your grade if you don't submit it. However, it should be good preparation for the exam.

Ungraded assignments:

For Wednesday, May 3:

1. Do §5.1, #3, 4. Doing all of these will be repetitive: do a sufficiently large collection.
2. Do §5.1, #6.
3. Do §5.1, #8.
4. Do §5.1, #9. What are the eigenvectors of these eigenvalues?
5. Do §5.1, #15.
6. Do §5.1, #17.
7. Read §5.2, up to p. 273.

For Monday, May 1:

1. Do the problems on this handout. Problem 3 should be preparation for Monday's lecture.
2. Read §5.1.
3. Do §5.1, #2 and part (i) of #3.
4. Do §5.1, #7.

For Friday, Apr. 28:

1. Do §4.3, #10—14. You will want to use Theorem 4.7, which we haven't proved yet in class.
2. Suppose that T : V → V is a linear transformation and that α and β are bases of V. Show that [T]_α^α and [T]_β^β have the same determinant.
3. Suppose that D : M_n×n(F) → F is an alternating, n-linear function. Let B be any n×n matrix with entries in F. Define E : M_n×n(F) → F by the formula E(A) = D(AB). Prove that E is alternating and n-linear. Use this to conclude that D(AB) D(I_n) = D(A) D(B) for any n×n matrices A and B with entries ini F.
4. Read §5.1.

For Wednesday, Apr. 26:

1. Do the exercises in this handout and ask at least one question about it. This should be good preparation for Wednesday's lecture.
2. Read §§4.3 and 4.4.
3. Do §4.3, #19, 21.
4. Do §4.3, #24.

For Friday, Apr 21:

1. Determine the dimension of the space Alt₂(F³, F) of alternating, 2-linear functions F³ × F³ → F. Can you predict a pattern for the dimension of Alt_m(Fⁿ, F) in terms of m and n? It might help to do a few more calculations.
2. Do §4.2, #23, 26, 30. You can use that the determinant is an alternating multilinear function in these.
3. Do §4.2, #27 without assuming that the determinant is an alternating multilinear function. We will discuss this in class as part of showing the determinant is alternating.
4. Read §4.3 and §4.4.

For Wednesday, Apr 19:

1. Write down the definition of an alternating m linear function from one vector space to another. Be careful that the book's notation uses a row convention and in class we used a column convention.
2. Do §4.5, #5, 6, 10.
3. Do §4.5, #15.
4. The following problem is a slightly more general version of §4.5, #17, 18: Let V and W be vector spaces. Prove that the set of alternating, m-linear functions from V to W (as defined in class) is a subspace of Func(V × ⋯ × V, W).
5. Prove that, for any field F, every alternating, 3-linear function F³ × F³ × F³ → F is a multiple of the determinant.
6. Determine the dimension of the space of alternating, 2-linear functions F³ × F³ → F.

For Monday, Apr 17:

1. Read §4.5. For Monday's lecture, it may also be helpful to review the proof of Theorem 4.3 on p. 212.
2. Do §4.2, #2—4.
3. Do a selection of §4.2, #5—22. It doesn't matter which you pick: the point is to get the techniques for calculating determinants into your fingers.

For Wednesday, Apr. 12:

1. Read §4.2.
2. Do §4.2, #1—12 (some of these were already assigned for Monday). This is a great set of exercises, because it goes through many of the most important properties of the determinant in a situation where the calculations are not too dificult to do.

For Monday, Apr. 10:

1. Read §4.1.
2. Do §4.1, #2—8.
3. Do §2.6, #6, 7.
4. Do §2.6, #19, 20.

For Friday, Apr. 7:

1. This question is repeated from last time: Let V be the vector space F². Let α be the standard basis of V and let β be the basis consisting of v₁ = (2,-1) and v₂ = (1,3). Let {1} be the standard basis of F¹. Compute [v_i^*]_α^{1} for i = 1, 2.
2. Do §2.6, #5, 10.
3. Do §2.6, #8, 19. These problems are closely related.

For Wednesday, Apr. 5:

1. Let V be the vector space F². Let α be the standard basis of V and let β be the basis consisting of v₁ = (2,-1) and v₂ = (1,3). Let {1} be the standard basis of F¹. Compute [v_i^*]_α^{1} for i = 1, 2.
2. Do §2.6, #2, 3, 4.

For Monday, Apr. 3:

1. Read §2.6.

For Monday, Mar. 24:

1. Read §2.6.
2. Do §3.4, #3. (I find it easier to do part (b) before part (a).)
3. Do §3.4, #5.
4. Do §3.4, #8.
5. Do §3.4, #12.
6. Do §2.6, #2,4.

For Wednesday, Mar. 22:

1. For each of the matrices A in §3.2, #2 and #5 do the following: find the rank of A; find the nullity of A; find a basis for the kernel of A; find a basis for the image of A; find equations for the image of A (in other words, find a matrix B whose kernel is the image of A).
2. Do §3.2, #20. Also find a 1×3 matrix N with rank 1 such that NA = O.
3. Do §3.3, #9, 10.
4. Do §3.4, #2, 9.

For Monday, Mar. 20:

1. Do §3.2, #5, 6.
2. Do §3.2, #14.
3. Do §3.2, #19, 21, 22.
4. Read §3.3 and §3.4.
5. Do §3.3, #2, 3.

For Friday, Mar. 17:

1. Study for the exam! The topic is linear transformations.

For Wednesday, Mar. 15:

1. Do §2.5, #4, 6, 7.
2. Do §2.5, #10. Ponder the following question: In our dictionary between linear transformations and matrices, what operation on linear transformation corresponds to the trace on matrices?
3. Do §2.5, #11. This should be very quick if you interpret the change of basis matrices correctly.
4. Do §2.5, #12. The point is really to understand the notation L_A and how it interacts with change of basis.
5. Each elementary row or column operation can be effected by multiplication by a matrix. Find these matrices. Do you multiply on the left or the right? (This is §3.1, #7.) Find the elementary row and column operations inverse to each of the elementary row and column operations.
6. Read §3.2.

For Monday, Mar. 13:

1. Do §2.4, #2.
2. Determine which of the matrices in §3.2, #5 are invertible and find their inverses by viewing a matrix M as [T]_α^β and find a new basis α' such that [T]_α'^β is the identity matrix.
3. In class, we have been developing a dictionary between linear transformations and matrices. Ponder the line of this dictionary that contains rank and nullity on the linear transformations side. If T : V → W is a linear transformation, can you find basis α and β for V and W in which it is easy to compute the nullity and rank of T?
4. Read §2.5.
5. Do §2.5, #2.

For Friday, Mar. 10:

1. Read §2.5 and §3.1.
2. There are three basic ways of changing a basis β = { v₁, ..., v_n } into a new basis. We could (i) exchange two vectors, (ii) scale one vector by a nonzero scalar, or (iii) replace v_i by v_i + cv_j for some scalar c and some index j ≠ i. Suppose that T : V → W is a linear transformation and that α and β are bases of V and of W. Find the effect on [T]_α^β of performing each of these operations to α or to β.
3. Do §2.4, #1, 3.
4. Prove Corollary 1 on p. 102. Your proof should be similar to the solutions of these problems advocated in class on Monday and Wednesday.
5. Do §2.4, #4, 5.
6. Do §2.4, #9. Prove as well that if T, U are linear transformations such that TU makes sense and is an isomorphism then both T and U are isomorphisms.
7. Do §2.4, #15. This proof was sketched in class on Wednesday.

For Wednesday, Mar. 8:

1. Read §2.4.
2. Do these problems (generalizing §2.3, #5). There are two ways to do the second problem, either by using the formula for matrix multiplication or by using corresponding facts for linear transformations, as discussed in class. Do it both ways.
3. Do §2.3, #9.
4. Do §2.3, #12.
5. Do §2.3, #13. Is it true that tr(ABC) = tr(CBA)?
6. Do §2.3, #15.

For Monday, Mar. 6:

1. Read §2.3 (again).
2. For each of the linear transformations on this handout find a basis on the vector space of linear plane motions and express your linear transformation in that basis. You will want to find a convenient basis for each of the linear transformations in question.
3. Do §2.2, #13.
4. Do §2.2, #16.
5. Do §2.3, #2, 3, 4.

For Friday, Mar. 3:

1. Suppose that V and W are vector spaces. Let ℒ(V,W) be the vector space of all linear transformations from V to W. Fix bases β of V and γ of W and let φ : ℒ(V,W) → M_m×n(F) be the function φ(T) = [T]_β^γ. Prove that φ is a linear transformation.
2. Read §2.3.
3. Do §2.2, #1.
4. Do §2.2, #9, 11.
5. Suppose that T : V → W is a linear transformation whose matrix in some bases of V and W is diagonal (that is, the (i,j)-th entry is zero whenever i≠j). Compute dim ker(T) and dim im(T) in terms of the diagonal entries of the matrix.

For Wednesday, Mar. 1:

1. Read §2.2.
2. Do §2.2, #2, 3, 5adg
3. Let β = { 1, x, x², ³ } be the standard basis of P₃(ℝ). Let f_i = ∏_{j ≠ i} (x - j) / (i - j) (the product is taken from j = 0 to j = 3) and let γ = { f₀, f₁, f₂, f₃ } be the Lagrange basis. Let T : P₃(ℝ) → P₃(ℝ) be the linear transformation T(f) = f' (the derivative). Find [T]_β^β, [T]_β^γ, [T]_γ^β, [T]_γ^γ.
4. Suppose that V is a vector space of dimension n and W is a vector space of dimension m. Compute the dimension of the vector space of linear transformations from V to W.
5. Suppose that T : V → W is a linear transformation that is bijective. Show that T^-1 : W → V is also a linear transformation.

For Wednesday, Feb. 22:

1. Let V be the subspace of M_3×3(ℝ) consisting of those matrices such that the sub of the entries along every row and every column is the same. Compute the dimension of V. This will be discussed in class.
2. §2.1, #17
3. §2.1, #19
4. §2.1, #21
5. Let V be the vector space of linear plane motions. Compute the image and kernel, and their dimensions, for each of the types of linear transformations we encountered on Friday.

For Monday, Feb. 20:

1. Read §2.1. The focus on Monday will be on applications of the dimension theorem (Theorem 2.3 in the book).
2. Prove that if T : ℝⁿ → ℝ^m is a linear transformation and L is a line in ℝⁿ then T(L) is either a line in ℝ^m or T(L) = { 0 }. I suggest using the concept of a parameterized line from §1.1.
3. Important: Prove that if T : V → W is a linear transformation and S ⊆ V is a linearly dependent set then T(S) ⊆ W is a linearly dependent set. Conclude that if S ⊆ V and T(S) is linearly independent then S is linearly independent.
4. Finish the problems handed out in class on Friday [pdf].
5. Do §2.1, #2—6. This time, compute the dimensions of the kernels and images by finding bases.
6. Do §2.1, #15 and #16.
7. Do §2.1, #28.
8. This is a variant of §2.1, #38. Let ℂ be the set of complex numbers. Show that the function f(x + iy) = x - iy is a linear transformation over the field ℝ but not over the field ℂ.

For Friday, Feb. 17:

1. Read §2.1.
2. Let V be the vector space of linear plane motions. Describe as many different kinds of linear transformations from V to itself as you can. We will discuss this in class on Friday.
3. Verify that the transformations defined in Examples 2, 3, and 4 on p. 66 are linear.
4. Do §2.1, #11 and #12.

For Wednesday, Feb. 15:

1. Read §2.1.
2. You may want to read §1.7 to see a construction of a basis for an infinite dimensional vector space. This is not required.
3. Do §1.6, #15—17
4. Suppose that W is a subspace of a vector space V and v is a vector in V. What are the possibilities for dim (W + span(v)?
5. Do §1.6, #28
6. On §2.1, #2—6, verify that the functions are linear.
7. Do §2.1, #9

For Monday, Feb. 13:

1. Read §1.6 (again, again, again); review the replacement theorem and the definition of dimension.
2. Do as many of these questions as you can.
3. Do §1.6, #10.
4. Do §1.6, #24.

For Friday, Feb. 10:

1. Read §1.6 (again, again).
2. Do §1.6, #18. What happens if you allows all sequences, as opposed to just those that have only finitely many nonzero entries?
3. Do §1.6, #10.
4. Do §1.6, #19. Can you find a way to generalize the statement for an arbitrary basis of an arbitrary vector space? Can you prove it?

For Wednesday, Feb. 8:

1. Read §1.6 (again).
2. Do §1.6, #8.
3. Do §1.6, #9.
4. In §1.6, #2, find the coordinates of the vector (1,1,1) in each of the subproblems where the vectors form a basis.
5. Do §1.6, #11.
6. Do §1.6, #13, 14.

For Monday, Feb. 6:

1. Read §1.6.
2. Do §1.6, #12.
3. Do §1.6, #17, 18.
4. Find a basis for ℂ as an ℝ-vector space and as an ℂ-vector space.
5. Is there any vector space V, over any field, such that the set of nonzero vectors of V is linearly independent?
6. Do §1.6, #19. Hint: use the fact that (x A + y B)^t = x A^t + y B^t.
7. Do §1.6, #2, 3. The calculations you have to do here are all things we have done before in our discussion of spans and linear independence, so it isn't important to go through every single calculation, although it is important to do a few. The really important thing is to make sure you can set up the systems of linear equations you need to solve, and that you know the significance of the solutions to those equations.

For Friday, Feb. 3:

1. Read §1.5 (again)
2. Do §1.5, #1
3. Do §1.5, #2, 3: You don't have to do all of these, but you should at least figure out how to translate each of them into a linear system of equations. You should go through the full calculation at least twice.
4. Do §1.5, #10

For Wednesday, Feb. 1:

1. §1.4, #11 (we discussed this in class), 13, 15
2. Read §1.5

For Monday, Jan. 30:

1. Read §1.4.
2. Do §1.3, #19 and 23. Please submit your solution to at least one of these problems on D2L as "Subspaces proof (ungraded)".
3. Do §1.3, #28
4. Do §1.4, #5—9

For Friday, Jan. 27:

1. Reread §1.2 and §1.3. Identify places where the book gives incomplete details, and make a list of the additional verifications the reader needs to do. Bring these to class on Friday.
2. Do §1.3, #7. Make sure that your proof uses the definitions in the text, and doesn't rely on intuition.
3. Do §1.3, #8, 10, 11.
4. Do §1.3, #17. This problem is closely related to our discussion at the end of class on Wednesday.
5. Do §1.3, #22 (compare to §1.2, #12).

For Wednesday, Jan. 25:

1. Submit a proof that something is a vector space. This is very important! I need to gauge your proofwriting skills and you need to get a sense of the requirements of the class. If you do not know which vector space to consider, show that ℝⁿ is a real vector space.
2. Revisit the problems assigned for Monday: If you did not know how to approach one of those problems then, can you figure out what to do now? If you did know how to do those problems, look back on them and see if you left out any important steps (such as checking that the operations are well-defined)? Look in particular at §1.2, #12.
3. Read §1.3. Submit questions about the reading!
4. Do §1.3, #8, #10.

For Monday, Jan. 23:

1. The list of problems below includes several that ask you to prove that a set with certain operations forms a vector space. You should do this in complete detail at least once, but it is okay to skip some details later on, as long as you think through what is involved in those details.
2. In order to get a sense of your proofwriting skills, I would like you to submit a proof that something is a vector space via D2L. You may want to submit one of the problems below, or to fill in the details of one of the examples from the textbook, or find another example, but please include all details. This assignment will not be graded, but I will return comments on your work.
3. Complete the LaTeX assignment. You may want to begin with this template, and you may want to use Overleaf to edit it. There are more LaTeX resources at the bottom of this page. You could combine the LaTeX assgnment with the proofwriting assignment...
4. §1.2, #2—4, 7
5. §1.2, #8: prove this carefully, showing all steps!
6. Prove that ℂ is an ℝ-vector space in which the zero vector 0 is the complex number 0 + 0i, vector addition is addition in the complex numbers ((a + bi) + (c + di) = (a+c) + (b+d)i), and scalar multiplication is multiplication in the complex numbers (c(a + bi) = (ca) + (cb)i).
7. Is the empty set a vector space?
8. Think of an example of a vector space (or an object that has some features of a vector space but not all of them) that is not mentioned in the book. Try to find examples that are not obviously mathematics.
9. Let V be the set ℝ². Define

   0 = (1,1)
   (x,y) + (z,w) = (x + z - 1, y + w - 1)
   c(x,y) = (cx - c + 1, cy - c + 1).

   Prove that, with these operations, V is a real vector space.
10. §1.2, #12
11. §1.2, #20

For Friday, Jan. 20:

1. Read §§1.1—1.2.
2. §1.1, #2: For each part, find a parameterization of the line (note that the textbook uses the term 'equation' instead of 'parameterization').
3. §1.1, #3: For each part, find a parameterization of the plane (note that the textbook uses the term 'equation' instead of 'parameterization').
4. §1.1, #6 and #7
5. Make up 3 questions about the reading assignment and submit them anonymously (you can also use the text box at the top of this page).