Colorado State University, Pueblo; Spring 2012
Math 307 — Introduction to Linear Algebra
Course Schedule & Homework Assignments

Here is a link back to the course syllabus/policy page.

In the following all sections and page numbers refer to the required course textbook, Linear Algebra, A Modern Introduction (2^nd edition), by David Poole.

This schedule is will be changing very frequently, please check it at least every class day, and before starting work on any assignment (in case the content of the assignment has changed).

• M:
  • Content:
    1. bureaucracy and introductions
    2. what is Linear Algebra (the study of vector spaces and linear transformations...)
    3. why do we do so much abstraction and formality at this point of the mathematics curriculum: what abstraction is good for
  • Miniquiz 0
• T:
  • Read: To the Student, p. xxiii and §§ 1.1–1.3
  • Miniquiz 1
  • Journals are not due today; it's a bit too early in the term. (First Journal entry will be due next Tuesday.)
  • Content:
    1. some basic terminology and notation:
       • logical and basic set theoretic terminology/notation
       • some basic sets of numbers
         1. natural numbers $\NN$
         2. integers $\ZZ$
         3. rationals $\QQ$
         4. real numbers $\RR$
       • starting good definitional style, including
         • all variables must be "bound"
         • clearly identify the symbol and/or terminology being defined
         • clearly identify the type of object being defined
       • vectors in $\RR^n$
       • vector addtion
       • scalar multiplication
       • the dot product
       • norms
    2. some basic properties
       • of vector arithmetic
       • of dot products and norms
       • the triangle inequality
  • Do HW0: Send me e-mail (to jonathan.poritz@gmail.com) telling me:
    1. Your name.
    2. Your e-mail address. (Please give me one that you actually check fairly frequently, since I may use it to contact you during the term.)
    3. Your year/program/major at CSUP.
    4. The reason you are taking this course.
    5. What you intend to do after CSUP, in so far as you have an idea.
    6. Past math classes you've had.
    7. Other math and science classes you are taking this term, and others you intend to take in coming terms.
    8. Your favorite mathematical subject.
    9. Your favorite mathematical result/theorem/technique/example/problem.
    10. Anything else you think I should know (disabilities, employment or other things that take a lot of time, etc.)
    17. [Optional:] If you were going to be trapped on a desert island alone for ten years, what music would you like to have?
• W:
  • Read: §§ 2.1 & 2.2
  • Miniquiz 2
  • Content:
    1. more basic terminology and notation...
       • complex numbers
       • angles between vectors
       • quantifiers $\forall$ and $\exists$
       • more on what makes a good defintion
       • [systems of] linear equations
       • solutions of linear systems, the solution set and its structure
       • a[n in]consistent linear system
       • the coefficient and augmented matrices of a linear system
       • elemenatry row operations (EROs)
       • row-equivalent matrices
       • [in]homogeneous linear systems
    2. more basic properties
       • solution sets of linear systems are either empty, have exactly one point, or have an infinite number of points
• F:
  • [Re]Read: §2.2
  • Content:
    1. yet more basic terminology and notation...
       • angles between vectors
       • orthogonal vectors
       • a [non]trivial solution of a linear system
       • relation between [non]trivial solutions, [non]homogeneous linear systems, and [non]unique solutions
       • more on what makes a good defintion
       • what makes a good statement of a result (theorem, propostion, etc.)
       • [starting on] what makes a good proof
       • proof structures:
         1. unpacking ("follow your nose")
         2. contradiction ("it can't not be true")
         3. many more to come...
       • the [reduced] row-echelon form of a matrix
       • row-reduction
       • free variables
       • the rank of a matrix
    2. more basic properties
       • The Rank-Nullity Theorem
  • Maxiquiz 1 handed out today, due on Monday
  • Today [Friday] is the last day to add classes.

• M:
  • [Re]Read: §2.2 and Read §2.3
  • Content:
    1. terminology we needed to say out loud:
       • Gaussian elimination
       • Gauss-Jordan elimination
    2. noticing that the definition we gave of the angle between two vectors needed $|\vec{v}\cdot\vec{w}| < 1$ in order to make sense; this is the content of the Cauchy-Scharz-Buniakowsky Inequality
    3. discussion: we say that some new term is well-defined if its definition makes sense, e.g.:
       • any formulæ in the definition can be successfully computed (as, for example, the Cauchy inequality shows we can compute the $\arccos$ in the definition of the angle between two vectors)
       • any value expressed in a definition must be clear, unambiguous, and unique (as, for example, the number of non-zero rows in the RRE version of a matrix is a uniquely defined number, which makes the rank of a matrix be well-defined)
    4. linear combination
    5. span
  • Hand in HW1: 1.2.62, 2.2.44, 2.2.47
  • Hand in Maxiquiz 1
  • Miniquiz 3
• T:
  • [Re]Read: §2.3
  • Content:
    1. going over Miniquiz 3, Maxiquiz 1 and HW1
    2. context and type for definitions ... see the handout on definitions for more on this theme
    3. some elementary facts about $\Span$ such as that each of the vectors $\vec{0},\vec{v}_1,\dots,\vec{v}_k$ is in the set $\Span(\vec{v}_1,\dots,\vec{v}_k)$.
    4. linearly [in]dependent vectors — note it is important that the scalars in the definition are not all zero
    5. examples of linearly [in]dependent vectors
    6. starting the discussion of the relationship between [in]dependence and matrix multiplication by a matrix whose columns are the vectors under consideration
  • Miniquiz 4
• W:
  • [Re]Read: §2.3
  • Content:
    1. more examples of linearly [in]dependent vectors
    2. proving vectors are linearly [in]dependent
    3. the relationship of linear [in]dependence and the linear system whose coefficient matrix has columns which are the vectors under consideration:
       • there will be a solution to the system if and only if the RHS vector is in the span of the columns
       • there will be a non-trivial solution of the homogeneous linear system with that coefficient if and only if the vectors are linearly dependent
  • Miniquiz 5
• F:
  • Read: §3.1
  • Content:
    1. proofs by induction (e.g., 3.1.37)
  • Hand in HW2: 2.3.43 and Chapter 2 Review problem 18 (on p.133)
  • Maxiquiz 2 today

• M:
  • [Re]Read: §3.1 and Read §3.2
  • Content:
    1. going over Maxiquiz 2 — the moral was write down the definitions, it's often enough!
    2. recall from Math 207 and your reading of the book the basic terminology:
       • matrices
       • matrix addition
       • scalar multiplication with matrices
       • matrix multiplication
       • the identity matrix
       • matrix algebra
    3. in class we recalled the definition of transpose
    4. starting the proof (by induction!) that the transpose of the sum of $k$ matrices is the sum of the transposes of those matrices, $\forall k$.
  • Miniquiz 6
  • Today [Monday] is the last day to drop classes without a grade being recorded
• T:
  • [Re]Read: §3.2 and Read §3.3
  • Content:
    1. finishing the induction proof that the transpose of the sum of $k$ matrices is the sum of the transposes of those matrices, for any $k$.
    2. how to submit electronic Journal entries and HWs.
    3. some work with matrix multiplication:
       • a very condensed version of the definition
       • remember: this operation is very rarely commutatitve
    4. stated without proof that the transpose of the product of $k$ square matrices is the product in the opposite order of the transposes of those matrices; noted that a proof of this would also use induction.
    5. [skew-]symmetric matrices, properties:
       • symmetric and skew-symmetric matrices must be square
       • a skew-symmetric matrix always has zeros on the diagonal
  • Hand in HW3: 3.1.37
  • Hand in (or submit electronically) your Journal 1 on §§1.1-1.3, 2.1, & 2.2 and Journal 2 on §§2.3, 3.1, & 3.2. See this link (or Blackboard) for more info on what is expected of you and how to do it.
  • Miniquiz 7
• W:
  • [Re]Read: §3.3
  • Content:
    1. going over HW2
       • style is important in proofs! ... Be stylish!
       • it helps to have a guess as to what you want to prove, then simply to write down all the definitions and see how they relate to each other (the unpacking strategy, after you decide what you want to unpack)
    2. a very logical day:
       • the details of an if–then statement, i.e., one in the form $P\Rightarrow Q$
       • the converse of $P\Rightarrow Q$ (which is $\neg P\Rightarrow\neg Q$).
       • the contrapositive of $P\Rightarrow Q$ (which is $\neg Q\Rightarrow\neg P$).
       • the negation of $P\Rightarrow Q$ (which is $P\land\neg Q$)
       • if $P\Rightarrow Q$ is true, the converse may or may not be true
       • an "if-then" statement is true if and only if its contrapositive is true
       • the negation of a statement of the form $\forall x\ P(x)$ is $\exists x\ \neg P(x)$
       • the negation of a statement of the form $\exists x\ P(x)$ is $\forall x\ \neg P(x)$
  • therefore, discussing what is the negation of the statement $\forall a,b,c\in\RR\ (a\vec{u}+b\vec{v}+c\vec{w}=\vec{0}\Rightarrow a=b=c=0)$
• F:
  • Hand in HW4: 3.2.33 and 3.3.46
  • Content:
    1. going over some recent HW and miniquizzes
    2. discussion of how thinking of and writing down proofs is a serious new skill we are working on in this class, so we should definitely not expect it to be terrifically easy and or to come quickly
    3. definition of the inverse of a matrix, and what it means for a matrix to be invertible
    4. proved the Theorem: Given $A$ and $B$ invertible matrices, $A\,B$ will be invertible, and $(A\,B)^{-1}=B^{-1}\,A^{-1}$.
  • Maxiquiz 3 today

• M:
  • Content:
    1. going over Maxiquiz 6
    2. review for Midterm I; see this review sheet
    3. discuss HW10 — you are responsible for these problems for tomorrow's midterm!
  • Journal 5 on §§4.1 & 4.3 is due today — show it to me or submit it electronically, but you can keep it to study from, if you like, and hand it in for real tomorrow
  • Miniquiz 15
• T:
  • Hand in Journal 5, if you are doing it on paper
  • Hand in HW10: 4.4.40-42, 4.4.47-48
  • Midterm I in class today
• W:
  • Read: §5.1
  • Content:
    1. going over Midterm I
    2. orthogonal sets of vectors in $\RR^n$
• F:
  • [Re}Read: §5.1
  • Content:
    1. orthogonal and linearly independent sets of vectors in $\RR^n$
    2. an orthonormal basis [ONB]
    3. coordinates with respect to an ONB
    4. orthogonal matrices:
       • definition
       • alternative characterizations
       • their effect on the dot product or the norm
       • the set of such is closed under products and inverses [Hence we call the set of $n\times n$ orthogonal matrices the orthogonal group and write it $O(n)$.]
       • determinants of orthogonal matrices
  • Hand in revised solutions to Midterm 1, if you like
  • Miniquiz 16