Welcome to **Honors Linear Algebra**! This is a rigorous, proof-based linear algebra class. It is different from a typical linear algebra course: we will cover many subjects in greater depth, and from a more abstract perspective. The expectations of this class are also higher, especially with respect to proof-writing and abstraction. The first thing to do when joining this course is to make sure these expectations align with your goals. If they do not, you might find a different linear algebra class a better fit.

This course has high expectations. You should plan to spend **9 hours per week** on this class, *not including lecture*. It will also be necessary to supply **independent motivation** as *not all of the work you need to do for this class will be collected, or even assigned*. It is also essential to **recognize early** when you are struggling with a concept and **discuss it with me**.

Above all, you must **engage actively** with the material as we learn it. If you are studying actively, you will have questions. Use this principle to measure whether you are actively engaged.

I encourage you to consult outside sources, use the internet, and collaborate with your peers. However, there are important rules to ensure that you use these opportunities in an academically honest way.

- Anything with your name on it must be your work and accurately reflect your understanding.
- Plagiarism will be dealt with harshly. Deliberate plagiarism will be reported to the Honor Code Office.

To avoid plagiarism, you should always **cite all resources you consult**, whether they are textbooks, tutors, websites, classmates, or any other form of assistance. Using others' words verbatim, without attribution, is absolutely forbidden, but so is using others' words with small modifications. *The ideal way to use a source is to study it, understand it, put it away, use your own words to express your newfound understanding, and then cite the source as an inspiration for your work.*

Do you have a question or comment about the course? The answer might be in the course policies, below. If your question isn't answered in the course policies, please send me an email. Or, if you prefer, you may send me a comment anonymously.

Instructor: Jonathan Wise

e-mail: jonathan.wise@colorado.edu

Office: Math 204

Office hours: calendar

Phone: 303 492 3018

My office is **Room 204** in the Math Department. My office hours often change, so I maintain a calendar showing the times I will be available. I am often in my office outside of those hours, and I'll be happy to answer questions if you drop by outside of office hours, provided I am not busy with something else. I am also happy to make an appointment if my office hours are not convenient for you.

There are many linear algebra textbooks, all of which have strengths and weaknesses. The official textbook for this course is:

Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence. *Linear Algebra*, 4e.

In case you have not been able to get a copy of the textbook yet, here are scans of the first two sections.

I plan to consult other textbooks to design the course, and you may want to consult them too. Here are a few that you may want to look at:

Sheldon Axler. *Linear Algebra done Right*, 3e.

Sergei Treil. *Linear Algebra done Wrong*.

Your final score in the class is based on your achievement in each of the major topics of the course. In each area, I will assign you a score between 0 and 5 based my perception of the depth of your understanding of that concept:

- To achieve a score of 1 (D), you should
- know and state the concept's definition precisely.
- To achieve a score of 2 (C), you should
- recognize examples, and
- perform computations correctly.
- To achieve a score of 3 (B), you should
- recognize situations where the concept applies, and
- set up computations using the concept.
- To achieve a score of 4 (A), you should:
- write formal proofs using the concept.
- To achieve a score of 5, you should:
- use the concept in sophisticated proofs.

Your score in each topic is based on the following kinds of work:

**Quizzes**: Every topic will be quizzed at least once in class and at least once on an exam.**Homework**: I will give a lengthy homework assignment on each topic. These can be revised.**Portfolios**: In a portfolio, you write a summary of the important parts of the topic, identify and complete relevant problems. Porfolios can be submitted via D2L or presented to me in my office.

Your score in each topic will be the *average of the two highest scores* from *different categories* above.

Your final score will be the *lowest* of your topic scores, adjusted upwards by up to a point according to your overall average. Here is the actual formula:

G = M + (A-M)/(4-M)

Here *G* is your final grade, *A* is the average of your topics scores, and *M* is the minimum of your topics scores. The number *G* is translated into a grade by a GPA scale.

- Writing (w)
- Can write coherent mathematical sentences.
- Can use the definitions correctly.
- Can write clear mathematical paragraphs using notation consistently; can reason about mathematical objects as presented.
- Can reason correctly, clearly, and consistently for the duration of a mathematical proof; can introduce concepts to aid the reasoning.
- Can coherent mathematical texts spanning multiple pages and applying linear algebra to topics that are not immediately recognizable as linear algebra.
- Vector spaces (vec): §§1.2, 1.3
- Be able to recite the definition of a vector space and a subspace.
- Perform algebraic operations correctly in vector spaces (using only the vector space operations); verify abstract identities concerning vectors in vector spaces.
- Understand how images, kernels, spans, solutions to linear equations produce linear subspaces; prove (or disprove) that subsets of ℝ
^{n}form subspaces. Recognize examples of vector spaces when you see them. - Prove that abstractly defined vector spaces are vector spaces (ℝ
^{n}, polynomials, linear functions, etc.); prove that kernel and image of a linear transformation are subspaces; construct the product of two vector spaces. - Construct the quotient of a vector space by a subspace.
- Basis and dimension (basis): §§1.5, 1.6
- Be able to recite the definitions of linear dependence, linear independence, generation, basis, dimension.
- Check for linear independence, generation, basis in ℝ
^{n}(e.g., using row reduction, determinants); compute the dimension of subspaces of ℝ^{n}. - Check for linear independence, generation, basis in subspaces of ℝ
^{n}and in general vector spaces. - Write rigorous proofs of linear independence, generation, basis, for a given collection of vectors.
- Prove that every vector space has a basis.
- Linear transformations (hom): §2
- Know the definition of a linear transformation, matrix multiplication, rank of a linear transformation (§3.2); multiply matrices (§2.3).
- Compute the rank of a matrix; determine if a matrix is invertible using determinant; compute the inverse of a matrix (§3.2).
- Find the matrix of a linear transformation with respect to a given basis (§2.2), find the matrix representation in one basis from the matrix representation in another basis (§2.5), find the matrices of abstractly presented linear transformations (projections, rotations, etc.); recognize linear transformations when you see them.
- Prove that the dual vector space is a vector space; prove that the space of linear transformations from one vector space to another is a vector space; prove that an abstractly presented linear transformation is linear.
- Formulate and prove the universal properties of subspaces and quotients.
- Kernel and image (ker)
- Know the definition of the kernel/null space, image/column space (§2.1), system of linear equations (§1.4).
- Find a basis for the kernel and the image using row/column operations; translate system of linear equations into questions about kernel and image of a linear transformation; find general solutions to systems of linear equations using row reduction and matrix inversion (§§1.4, 3.1, 3.2).
- Understand the relationship between dimension of the kernel, dimension of the image, and dimension of the domain; know criteria for invertibility of a linear transformation; know the effect of row and column operations on image and kernel.
- Calculate kernels, images, and their dimensions in abstract settings, with justification.
- Dual vector space and cokernel (dual): §2.6
- Recite the definition of the dual vector space (§2.6), transpose of a matrix.
- Find a basis for the dual subspace of a subspace of ℝ
^{n}. - Recite the definition of the quotient of a vector space by a subspace; compute the dimension of the quotient of a vector space by a subspace.
- Prove that the quotient of a vector space by a subspace is a vector space.
- Construct an isomorphism between the quotient and the dual of the kernel of the transpose.
- Determinant (det): §4
- How to calculate by row operations, row/column expansion; cross product
- Relation to volume, orientation; solve system of linear equations using Cramer's rule (§4.3); compute the determinant of a square matrix.
- Know the effects of inversion, transpose, composition, row operations on the determinant and use these in computation (§4.3); compute the determinant of a linear operator on an abstract vector space.
- Write proofs using determinants, with justification; use determinants to analyze invertibility, rank of matrices.
- Universal alternating function (§4.5)
- Eigenvalues and eigenvectors (eigen): §5
- Be able to recite the definitions of eigenvalues, eigenvectors, characteristic polynomial (§5.1), state the Cayley-Hamilton theorem (§5.4).
- Find eigenvalues of a matrix using characteristic polynomial, find the eigenvectors of an eigenvalue of a matrix (§5.1)
- Determine whether a matrix is diagonalizable, and find a basis in which it is diagonal (§5.1)
- Find and prove a closed formula for the Fibonacci sequence (or similar recurrences) using eigenvalues and eigenvectors.
- Prove the Cayley-Hamilton theorem (§5.4).

Time permitting, we may cover a few more topics. Should we have time for these topics, I will fill out the syllabus for them during spring break.

- Multilinear algebra (multi): §6.1
*(time permitting)* - Recite the definition of a bilinear form, symmetric bilinar form, positive definitie bilinar form (§6.1).
- Apply the Gram-Schmidt process to find an orthonormal basis of a symmetric bilinear form (§6.2).
- Apply the Gram-Schmidt process in abstract vector spaces.
- Jordan canonical form (jordan): §7
- Know the shape of a Jordan canonical form (§7.1).
*(time permitting)*

There will be daily, uncollected homework assignments. You are **expected to do and fully understand these assignments**, and to ask me for comments about your solutions if you are unsure of them.

There will also be collected homework assignments, due roughly once every two weeks. All submitted assignments should be **typeset legibly**, preferably using LaTeX, and uploaded using D2L **in PDF format**.

Unless otherwise indicated, **all reading assignments and exercises are from Friedberg, Insel, and Spence**.

- 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.

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

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

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

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

- Determine the dimension of the space Alt
_{2}(**F**^{3},**F**) of alternating, 2-linear functions**F**^{3}×**F**^{3}→**F**. Can you predict a pattern for the dimension of Alt_{m}(**F**^{n},**F**) in terms of*m*and*n*? It might help to do a few more calculations. - Do §4.2, #23, 26, 30. You can use that the determinant is an alternating multilinear function in these.
- 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.
- Read §4.3 and §4.4.

- 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. - Do §4.5, #5, 6, 10.
- Do §4.5, #15.
- 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*). - Prove that, for any field
**F**, every alternating, 3-linear function**F**^{3}×**F**^{3}×**F**^{3}→**F**is a multiple of the determinant. - Determine the dimension of the space of alternating, 2-linear functions
**F**^{3}×**F**^{3}→**F**.

- Read §4.5. For Monday's lecture, it may also be helpful to review the proof of Theorem 4.3 on p. 212.
- Do §4.2, #2—4.
- 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.

- Read §4.2.
- 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.

- Read §4.1.
- Do §4.1, #2—8.
- Do §2.6, #6, 7.
- Do §2.6, #19, 20.

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

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

- Read §2.6.

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

- 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*). - Do §3.2, #20. Also find a 1×3 matrix
*N*with rank 1 such that*NA = O*. - Do §3.3, #9, 10.
- Do §3.4, #2, 9.

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

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

- Do §2.5, #4, 6, 7.
- 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?
- Do §2.5, #11. This should be very quick if you interpret the change of basis matrices correctly.
- Do §2.5, #12. The point is really to understand the notation
*L*and how it interacts with change of basis._{A} - 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.
- Read §3.2.

- Do §2.4, #2.
- Determine which of the matrices in §3.2, #5 are invertible and find their inverses by viewing a matrix
*M*as*[T]*and find a new basis α' such that_{α}^{β}*[T]*is the identity matrix._{α'}^{β} - 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*? - Read §2.5.
- Do §2.5, #2.

- Read §2.5 and §3.1.
- There are three basic ways of changing a basis β = {
**v**_{1}, ...,**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}+ c**v**_{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]*of performing each of these operations to α or to β._{α}^{β} - Do §2.4, #1, 3.
- Prove Corollary 1 on p. 102. Your proof should be similar to the solutions of these problems advocated in class on Monday and Wednesday.
- Do §2.4, #4, 5.
- 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. - Do §2.4, #15. This proof was sketched in class on Wednesday.

- Read §2.4.
- 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.
- Do §2.3, #9.
- Do §2.3, #12.
- Do §2.3, #13. Is it true that
*tr(ABC) = tr(CBA)*? - Do §2.3, #15.

- Read §2.3 (again).
- 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.
- Do §2.2, #13.
- Do §2.2, #16.
- Do §2.3, #2, 3, 4.

- 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*be the function_{m×n}(F)*φ(T) = [T]*. Prove that φ is a linear transformation._{β}^{γ} - Read §2.3.
- Do §2.2, #1.
- Do §2.2, #9, 11.
- 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.

- Read §2.2.
- Do §2.2, #2, 3, 5adg
- Let
*β = { 1, x, x*be the standard basis of^{2},^{3}}*P*. Let_{3}(ℝ)*f_i = ∏*(the product is taken from_{j ≠ i}(x - j) / (i - j)*j = 0*to*j = 3*) and let*γ = { f*be the Lagrange basis. Let_{0}, f_{1}, f_{2}, f_{3}}*T : P*be the linear transformation_{3}(ℝ) → P_{3}(ℝ)*T(f) = f'*(the derivative). Find*[T]*._{β}^{β}, [T]_{β}^{γ}, [T]_{γ}^{β}, [T]_{γ}^{γ} - 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*. - Suppose that
*T : V → W*is a linear transformation that is bijective. Show that*T*is also a linear transformation.^{-1}: W → V

- Let
*V*be the subspace of*M*consisting of those matrices such that the sub of the entries along every row and every column is the same. Compute the dimension of_{3×3}(ℝ)*V*. This will be discussed in class. - §2.1, #17
- §2.1, #19
- §2.1, #21
- 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.

- Read §2.1. The focus on Monday will be on applications of the dimension theorem (Theorem 2.3 in the book).
- Prove that if
*T : ℝ*is a linear transformation and^{n}→ ℝ^{m}*L*is a line in ℝ^{n}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. **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.- Finish the problems handed out in class on Friday [pdf].
- Do §2.1, #2—6. This time, compute the dimensions of the kernels and images by finding bases.
- Do §2.1, #15 and #16.
- Do §2.1, #28.
- 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.

- Read §2.1.
- 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. - Verify that the transformations defined in Examples 2, 3, and 4 on p. 66 are linear.
- Do §2.1, #11 and #12.

- Read §2.1.
- You may want to read §1.7 to see a construction of a basis for an infinite dimensional vector space. This is not required.
- Do §1.6, #15—17
- 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**)? - Do §1.6, #28
- On §2.1, #2—6, verify that the functions are linear.
- Do §2.1, #9

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

- Read §1.6 (again, again).
- Do §1.6, #18. What happens if you allows
*all*sequences, as opposed to just those that have only finitely many nonzero entries? - Do §1.6, #10.
- 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?

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

- Read §1.6.
- Do §1.6, #12.
- Do §1.6, #17, 18.
- Find a basis for ℂ as an ℝ-vector space and as an ℂ-vector space.
- Is there any vector space
*V*, over any field, such that the set of nonzero vectors of*V*is linearly independent? - Do §1.6, #19. Hint: use the fact that
*(x A + y B)*=^{t}*x A*.^{t}+ y B^{t} - 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.

- Read §1.5 (again)
- Do §1.5, #1
- 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.
- Do §1.5, #10

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

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

- 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.** - Do §1.3, #7. Make sure that your proof uses the
*definitions in the text*, and doesn't rely on intuition. - Do §1.3, #8, 10, 11.
- Do §1.3, #17. This problem is closely related to our discussion at the end of class on Wednesday.
- Do §1.3, #22 (compare to §1.2, #12).

**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 ℝ^{n}is a real vector space.- 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.
- Read §1.3.
**Submit**questions about the reading! - Do §1.3, #8, #10.

- 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.
- 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. - 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...
- §1.2, #2—4, 7
- §1.2, #8: prove this carefully, showing all steps!
- 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*). - Is the empty set a vector space?
- 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. - Let
*V*be the set ℝ^{2}. Define**0**=*(1,1)*

*(x,y) + (z,w) = (x + z - 1, y + w - 1)*

*c(x,y) = (cx - c + 1, cy - c + 1)*.*V*is a real vector space. - §1.2, #12
- §1.2, #20

- Read §§1.1—1.2.
- §1.1, #2: For each part, find a parameterization of the line (note that the textbook uses the term 'equation' instead of 'parameterization').
- §1.1, #3: For each part, find a parameterization of the plane (note that the textbook uses the term 'equation' instead of 'parameterization').
- §1.1, #6 and #7
- 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).

Monday, March 20

Wednesday, February 22

Monday, February 20

Linear operations on linear plane motions from Friday, February 17

Replacement theorem from Monday, February 13

Activity from Friday, January 27

Wednesday, January 25

Monday, January 23

Friday, January 20

Wednesday, January 18

Exam 1: Friday, February 24, in class

Exam 2: Friday, March 17, in class

Exam 3: Friday, April 14, in class

A reference discussing most of the important aspects of LaTeX you will use.

A very quick introduction to LaTeX.

If you need to know the command for some symbol, try using Detexify.

Agnès Beaudry and Kate Stange list more LaTeX resources...

The Mathematics Academic Resource Center (MARC) is staffed with learning assistants and undergraduate and graduate students that can help you with concepts in this class. This is an excellent resource, and I encourage you to use it, but **remember to use this resource responsibly!**

**Do** ask for help from MARC with the daily, uncollected homework assignments in this class.

**Do** Ask for clarification of ideas from the textbook and from discussion in class.

**Do not** ask for help from MARC with specific problems on the the larger, collected (graded) assignments.

**Do not under any circumstances** submit a solution that was given to you by MARC as your own work.

The Office of Academic Affairs officially recommends a number of statements for course syllabi, all of which are fully supported in this class.

If you need special acommodation of any kind in this class, or are uncomfortable in the class for any reason, please contact me and I will do my best to remedy the situation. You may contact me in person or send me an anonymous comment.