Colorado State University, Pueblo; Fall 2013
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 (3^rd edition), by David Poole.

Also in the following, the image means that class was videoed that day and can be seen through the Blackboard page for this class. Note that if you know ahead of time that you will miss a class, you should tell me and I will be sure to video that day.

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
    4. an example of an application of linear algebra: the \$282 billion eigenvalue problem: Google's PageRank algorithm
  • Miniquiz 0
• T:
  • Read: To the Student, p. xxv&xxvi and §§ 1.1&1.2
  • 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. logical and basic set theoretic terminology/notation
    2. some basic sets of numbers
       1. natural numbers $\NN$
       2. integers $\ZZ$
       3. rationals $\QQ$
       4. real numbers $\RR$
    3. 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
    4. vectors in $\RR^n$
    5. vector addtion
    6. scalar multiplication
    7. the dot product
    8. norms
    9. linear combinations of vectors
    10. 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:
  • [Re]Read: §§ 1.1 & 1.2
  • Miniquiz 2
  • Content:
    1. the Cauchy-Schwartz Inequality
    2. angles between vectors
    3. orthogonal vectors
    4. unit vectors
    5. distances between vectors
    6. projection of one vector onto another
• F:
  • [Re]Read: §§ 1.1-1.3 & 2.1
  • Content:
    1. proof that the difference between a vector and its projection onto another vector is orthogonal to that second vector
    2. quantifiers $\forall$ and $\exists$
    3. [systems of] linear equations
    4. solutions of linear systems, the solution set and its structure
    5. [in]consistent linear system
    6. the coefficient and augmented matrices of a linear system
  • Maxiquiz 1
  • Today [Friday] is the last day to add classes.

• M:
  • Yes, we do have class today, even though it is the federal Labor Day holiday.
  • [Re]Read: §2.1 and Read §2.3
  • Content:
    1. going over Maxiquiz 1
    2. notation $A\Rightarrow B$ meaning "if $A$ then $B$" and $A\Leftrightarrow B$ meaning "$A$ if and only if $B$", also written "$A$ iff $B$"
    3. [in]homogeneous linear system
    4. relationship between a linear system being homogeneous and being consistent
    5. linear combination [again]
    6. Span, a few basic examples and properties (e.g., the span of a single vector is the line along that vector)
    7. (size of) solution sets of a linear system: empty, single vector, or an infinite number of vectors.
    8. result that a linear system is consistent iff the vector consisting of the right hand side constant values is in the span of the columns
  • Hand in HW1: 1.2.62, 2.2.44, 2.2.47
  • Miniquiz 3
• T:
  • [Re]Read: §2.3
  • Content:
    1. [non]trivial for both linear combinations and solutions of a linear system
    2. some elementary facts about $\Span$ such as that each of the vectors $\vec{0},\vec{v}_1,\dots,\vec{v}_k$, and the zero vector $\vec{0}$ are in the set $\Span(\vec{v}_1,\dots,\vec{v}_k)$.
    3. linearly [in]dependent vectors — note it is important that the scalars in the definition are not all zero
    4. examples of linearly [in]dependent vectors, including:
       • the set $\{\vec{0}\}$ is linearly dependent
       • if $\{\vec{u},\vec{v}\}$ is a linearly dependent set, then $\vec{v} = \alpha\,\vec{u}$ for some $\alpha\in\RR$.
       • if $\{\vec{v_1},\dots,\vec{v_k}\}$ is a linearly dependent set, then one of the $\vec{v_j}$ is linear combination of the remaining vectors.
       • proof (by contradition) that if $\{\vec{v_1},\dots,\vec{v_k}\}$ is a linearly independent set, then any subset is also linearly independent.
  • Miniquiz 4
  • hand in Journal 1 on material of Chapter 1 and the first part of Chapter 2
• W:
  • [Re]Read: §2.3
  • Content:
    1. more on proofs by contradiction
       • general remarks
       • a very classical example: Euclid's proof of the infinitude of the primes
    2. revisit Span a bit
       • it is a set
       • ... work at the level of sets, "anything you can do, I can do meta", as Daniel Dennett said to Douglas Hofstadter.
    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 non-trivial solution of the homogeneous linear system with that coefficient matrix if and only if the vectors are linearly dependent
  • Miniquiz 5
• F:
  • Read: §2.3
  • Content:
    1. rank, free variables, row reduction, etc.
  • Hand in HW2: 2.3.44 and Chapter 2 Review problem 18 (on p.141)
  • Maxiquiz 2 today

• M:
  • [Re]Read: §2.3
  • Content:
    1. going over Maxiquiz 2 — the moral was write down the definitions, it's often enough!
    2. rank
    3. a term is well-defined if any choices in its definition are mentioned explicitly
    4. rank is well defined because matrices always do have an RREF form, and that form is unique; they also always do have an REF form, but that is not unique.
    5. necessary linear dependence of $k$ vectors in $\RR^n$ if $k>n$
  • Miniquiz 6
  • Today [Monday] is the last day to drop classes without a grade being recorded
• T:
  • Read: §3.1 and Read §3.2
  • Content:
    1. context and type for definitions ... see the handout on definitions for more on this theme
    2. going over recent miniquizzes and homework, noticed:
       • It often pays to write a scratch version of a proof before putting down the official version on a quiz or HW.
       • Students are sometimes (often?) forgetting to do both directions of the proof of an "if and only if" statement.
       • When you are trying to prove that two sets are equal, $S=T$, often the best way is to take a general element $s\in S$, write it down specifically (yes, a specific form of a general element!), and show that this particular element must also be in $T$, thereby showing all of $S$ is in $T$, i.e., $S\subseteq T$. Then you turn around and do the same thing for an element of $T$, getting $T\subseteq S$. It then follows that $S=T$.
    3. the converse of $P\Rightarrow Q$, which is $Q\Rightarrow P$) (or, equivalently, $\neg P\Rightarrow\neg Q$)
    4. the contrapositive of $P\Rightarrow Q$, which is $\neg Q\Rightarrow\neg P$
    5. if $P\Rightarrow Q$ is true, the converse may or may not be true
    6. an "if-then" statement is true if and only if its contrapositive is true
    7. definition of the inverse of a matrix, and what it means for a matrix to be invertible
    8. We talked about the method of proof by induction, or proofs using the Principle of Mathematical Induction, which goes like this:
       • It only applies to theorems of the specific form "$\forall n\in\NN\ S(n)$ is true," where $S(n)$ is a mathematical statement which depends upon a natural number parameter $n$.
       • First one proves that $S(1)$ is true; this is called the base case.
       • Then one proves "If $S(n)$, then $S(n+1)$"; this is called the inductive step and, during the proof of this step, when one invokes the hypothesis $S(n)$, one calls it the inductive hypothesis.
       • One declares the theorem proven by induction (and goes home happy).
    9. an example of an inductive proofs, to show that $\sum_{j=1}^n j = \frac{n(n+1)}{2}$.
    10. another example: proving that $\forall n\in\NN$, if $A_1,\dots,A_n$ are invertible matrices, then $\left(A_1\cdot\dots\cdot A_n\right)^{-1}=A_n^{-1}\cdot\dots\cdot A_1^{-1}$.
    11. ended class with the inductive proof that all pigs are yellow. [see if you can find the flaw in that proof.]
  • Hand in Journal 2
  • Miniquiz 7 (handed out)
• W:
  • [Re]Read: §§3.1-3.3
  • Content:
    1. 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
    2. in class we recalled the definition of transpose
    3. [skew-]symmetric matrices, properties:
       • symmetric and skew-symmetric matrices must be square
       • a skew-symmetric matrix always has zeros on the diagonal
    4. some more logic:
       • 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 $\forall x\ \neg P(x)$
    5. gave the book's definition of a subspace of $\RR^n$: it is a subset $V\subseteq\RR^n$ satisfying the properties:
       1. $\vec{0}\in S$
       2. $\forall \vec{u},\vec{v}\in S\ \vec{u}+\vec{v}\in S$
       3. $\forall \vec{u}\in S,\forall\alpha\in\RR\ \alpha\vec{u}\in S$
    6. examples of subspaces of $\RR^2$ and $\RR^3$:
       1. the trivial subspace $\left\{\vec{0}\right\}$
       2. any line through the origin
       3. any plane through the origin
       4. the whole thing ($\RR^2$ or $\RR^3$ as a subspace of itself)
  • Miniquiz 8
• F:
  • Read: §3.5
  • Hand in HW3: 3.1.37, 3.2.33 and 3.3.46
  • Content:
    1. going over some recent HW and miniquizzes
    2. discussed the flaw with the induction proof of the (false) theorem that all pigs are yellow: the inductive step did not prove "$\forall n\in\NN S(n)\Rightarrow S(n+1)$", because it did not work unless $n\ge 3$; in effect it proved that "$\forall n\in\NN,\ n\ge 3\land S(n)\Rightarrow S(n+1)$". That different inductive step would have proven the theorem "$\forall n\in\NN,\ n\ge 3, S(n)$" if we had used the base case $S(3)$. Instead, using the base case $S(1)$, as we did, the induction doesn't even get started and no theorem at all is proven.
    3. thinking more about subspaces of $\RR^n$... Notice that the first part of the book's definition is actually not necessary as long as the subset $S$ is non-empty: if it has any vector $\vec{u}$ at all, then it has $0\cdot\vec{u}=\vec{0}$ as well. And the second and third parts of the book's definition are together saying that subspaces are closed under linear combinations.
       So here is another, equivalent definition of what it means for a subset $S\subseteq\RR^n$ to be a subspace:
       1. $S\neq\emptyset$ (remember the notation for the empty set)
       2. $\forall \vec{u},\vec{v}\in S,\forall\alpha,\beta\in\RR\ \ \alpha\vec{u}+\beta\vec{v}\in S$
    4. building the list of examples of subspaces of $\RR^n$:
       1. the trivial subspace $\left\{\vec{0}\right\}$
       2. any line through the origin
       3. any plane through the origin
       4. a hyperplane through the origin in higher dimensions (e.g., the set of vectors $\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix}\in\RR^4$ with components satisfying $a\,x_1+b\,x_2+c\,x_3+d\,x_4=0$, where $a,b,c,d\in\RR$, is a "three-dimensional hyperplane" in $\RR^4$ (think of it as a linear system with only one equation: it has three free variables, so three parameters are needed to specify a point on this hyperplane).
       5. $\RR^n$ is itself a subspace of $\RR^n$. (Note that any other subspace than this one, so any subspace of $\RR^n$ which is not all of $\RR^n$, is called a proper subspace.)
       6. Spans are subspaces: $\forall n,k\in\NN$ and $\forall\vec{v}_1,\dots,\vec{v}_k\in\RR^n$, $\Span(\vec{v}_1,\dots,\vec{v_k})$ is a subspace of $\RR^n$
    5. so the span of a bunch of vectors is a subspace, but it need not be an efficient way to describe that subspace. Looking for a way to characterize an efficient set of vectors to build a subset, we defined a basis of a subspace of $\RR^n$
    6. examples of bases:
       1. the trivial subspace does not have a basis
       2. the standard basis of $\RR^n$ (which we have met before; it is the $n$ vectors $$\vec{e}_1=\begin{pmatrix}1\\0\\0\\\vdots\\0\\0\end{pmatrix},\vec{e}_2=\begin{pmatrix}0\\1\\0\\\vdots\\0\\0\end{pmatrix},\dots,\vec{e}_n=\begin{pmatrix}0\\0\\0\\\vdots\\0\\1\end{pmatrix},$$ where $\vec{e}_j$ is the vector in $\RR^n$ which has a $1$ in the $j^\text{th}$ component and $0$'s everywhere else; the dimension $n$ is not part of the notation $\vec{e}_j$, it must be understood from context) is, as the name suggests, a basis.
  • Maxiquiz 3 was handed out today, and you can hand it in on Monday, if you like, but it is sufficiently similar to recent homework problems that skipping that would not be a problem.

• M:
  • [Re]Read: §3.5
  • Content:
    1. proved the Theorem: Given $A$ and $B$ invertible matrices, $A\,B$ will be invertible, and $(A\,B)^{-1}=B^{-1}\,A^{-1}$.
    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; also, it is a very good idea to read the proofs in the book, they will be a source of inspiration for your own proofs in the future.
    3. going over HW3 — some of the morals were:
       • "repeat until terms are gone..." really means induction, which would be better to write out explicitly
       • when you do induction, you must:
         1. clearly write down the statement $S(n)$ over which you are inducting -- the hint will be if the thing to be proven has the structure $\forall n\in\NN\ S(n)$.
         2. make sure you do a reasonable base case
         3. clearly enunciate the logic of each step, with lots of transition words/phrases
       • be careful with induction, if you are not, you can make mistakes — e.g., the famous proof that all pigs are yellow [which is unfortunately incorrect!] — so your best bet is to stick carefully to the standard structure of induction proofs and to check over each step very carefully
    4. recalling the idea of a basis of a subspace of $\RR^n$, we noticed that a subspace which is a line (through the origin, of course), has many bases — any single, non-zero vector on the line will be a basis.
    5. likewise, a subspace consisting of a plane (through the origin in $\RR^n) has many bases: any two non-zero vectors which are not mulitples of each other
    6. therefore bases are very much not uniquely determined by their subspaces ... but the number of vectors in a basis does seem to be uniquely determined by the subspace; hence we define the dimension of a subspace of $\RR^n$
    7. "Dimension" is well-defined, says The Basis Theorem: Given a subspace $S$ of $\RR^n$, any two bases of $S$ have the same number of vectors.
    8. Started the proof of The Basis Theorem.
  • Miniquiz 9
• T:
  • [Re]Read: §3.5
  • Content:
    1. mentioned again that good definitions have all the components described in this guide
    2. finished the proof of The Basis Theorem.
    3. notice all subspaces of $\RR^n$ have either exactly one vector in them (in which case we're talking about the trivial subspace), or an infinite number of vectors
    4. to measure the size of a subspace, we defined the dimension of a subspace of $\RR^n$
    5. defined the row space and column space of a matrix
  • Hand in Journal 3
  • Miniquiz 10
• W:
  • [Re]Read: §3.5
  • Content:
    1. noticed that the previously defined row and column spaces are subspaces (of $\RR^n$ and $\RR^m$, respectively, for an $m\times n$ matrix)
    2. defined the null space of an $m\times n$ matrix, and proved it is a subspace of $\RR^n$
    3. from the row space comes the rank
    4. from the null space comes the nullity
    5. The Rank-Nullity Theorem
    6. notice that the row space of a matrix doesn't change as we do EROs to the matrix — the column space does change
    7. a basis of the row space of a matrix will consist of the non-zero rows of its RREF form.
  • Miniquiz 11
• F:
  • [Re]Read: §3.5
  • Content:
    1. more on the proof of The Rank[-Nullity] Theorem
    2. finding bases of the null space, column space and row space
    3. a part of The Fundamental Theorem of Invertible Matrices
    4. notation: TFAE = "the following are equivalent"; i.e., the following statements are all joined by "iff"
  • Hand in HW4: 3.5.4, 3.5.34, 3.5.62
  • Maxiquiz 4 today

• M:
  • Read: §3.6
  • Content:
    1. going over Maxiquiz 4 and HW4 in some detail
    2. defining a linear transformation
  • Miniquiz 12
• T:
  • [Re]Read: §3.6
  • Content:
    1. examples of linear transformations:
       • the trivial transformation which sends every input to $\vec{0}$.
       • the identity transformation $T:\RR^n\to\RR^n$ in any dimension $n\in\NN$
       • linear transformations $T:\RR^1\to\RR^1$:
         • not the same thing as functions whose graphs are a line
         • any such $T$ given by $T((x))=(m\cdot x)$, where $m\in\RR$, (thinking of $(x)$ as a vector of length one) is indeed a linear transformation — so the problem with the general function whose graph is a line is its $y$-intercept
         • in fact, all linear transformations $T:\RR^1\to\RR^1$ are given by such $T((x))=(m\cdot x)$, where $m\in\RR$. $m=1$ gives the identity transformation, $m=0$ gives the trivial transformation
       • linear transformations $T:\RR^1\to\RR^1$:
         • thinking about the geometry made us think that rotations, reflections, and dilations should be linear transformations; unclear about translations
         • by looking at what happened to unit vectors on the axes and assuming the rotation was linear and so could be extended to all other vectors using linear combinations, we got the matrix $R_\theta=\begin{pmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{pmatrix}$ to represent a counterclockwise rotation of $\RR^2$ by the angle $\theta$
         • similarly, a reflection of $\RR^2$ across the $y$-axis is represented by the matrix $F_y=\begin{pmatrix}-1&0\\ 0&1\end{pmatrix}$ and across the $x$-axis by $F_x=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$ .
         • dilation by scale factor $k$ is represented by the matrix $D_k=\begin{pmatrix}k&0\\ 0&k\end{pmatrix}$ .
         • in each case, when we say a linear transformation is "represented by a matrix", this is in the sense of the next theorem
       • Theorem If $A$ is an $m\times n$ matrix for some $m,n\in\NN$, then the map $L_A:\RR^n\to\RR^m$ given by $L_A(\vec{x})=A\cdot\vec{x}$ is a linear transformation.
       • It is left to see if the converse of that theorem is true, and how composition of linear transformations can be described in terms of their matrices.
  • Journal 4 (on everything up to and including §3.5) is due today
  • Miniquiz 13
• W:
  • [Re]Read: §3.6
  • Content:
    1. a Proposition: If $f:\RR^n\to\RR^m$ is a linear transformation, then $f(\vec{0})=\vec{0}$.
    2. as a consequence, a translation — the map $T_\vec{a}:\RR^2\to\RR^2$ given by $T_\vec{a}(\vec{x})=\vec{x}+\vec{a}$ for some fixed $\vec{a}\in\RR^2$ — is not a linear transformation
    3. more examples of linear transformations: projections
    4. a linear transformation coming from matrix multiplication — what the book (and almost no one else) calls a "matrix transformation"
    5. a linear transformation is determined by what it does to the standard basis
    6. the matrix $[T]$ of a linear transformation $T$, with examples:
       • projection onto the $x$-axis in $\RR^2$ has matrix $\begin{pmatrix}1&0\\ 0&0\end{pmatrix}$ .
    7. the composition of linear transformations has matrix which is the product of the matrices of the constituent transformations
  • Miniquiz 14
• F:
  • [Re]Read: §3.6
  • Content:
    1. repeating the Theorem that the composition of linear transformations is also a linear transformation and its associated matrix is the product (in the appropriate order) of the matrices associated to the constituent transformations
    2. proved the Proposition that given two linear transformations $S,T:\RR^n\to\RR^m$, the transformation $S+T$, defined as $(S+T)(\vec{v})=S(\vec{v})+T(\vec{v})$, is also a linear transformation
    3. domain, codomain, and range of a linear transformation
    4. recall the definition of onto (that the range equals the codomain); examples of linear transformations we know that are either onto or not
    5. recall the defintion of 1-1 (that $T(\vec{u})=T(\vec{v})$ happens only when $\vec{u}=\vec{v}$)
    6. Theorem: A linear transformation is 1-1 iff and only if its associated matrix has nullity zero. In symbols: a linear transformation $T:\RR^n\to\RR^m$ is 1-1 iff $\operatorname{nullity}([T])=0$.
    7. Proposition: A linear transformation $T:\RR^n\to\RR^m$ cannot be 1-1 if $n>m$.
    8. Corollary: If a linear transformation $T:\RR^n\to\RR^m$ is invertible, then $n=m$.
  • Maxiquiz 5 today
  • Hand in HW5: 3.6.4, 3.6.8, 3.6.44, 3.6.54

• M:
  • Read: §4.2
  • Content:
    1. going over Maxiquiz 5 and HW5
    2. review of some of the parts of the The Fundamental Theorem of Invertible Matrices, particularly the fact that a matrix $A$ is invertible iff it's RREF form is the $n\times n$ identity.
    3. the inverse of a linear/matrix transformation
    4. definition of the determinant
       • for $1\times 1$ matrices
       • for $2\times 2$ matrices
       • recursively for $n\times n$ matrices, where $n\ge 2$ — which is essentially Laplace's Expansion Theorem
    5. facts about determinants:
       • the determinant determines if a matrix is invertible: $\det(A)=0\ \Leftrightarrow\ A$ is invertible.
       • $\forall A,B,\ \det(AB)=\det(A)\cdot\det(B)$ — which is amazing, because matrix multiplication is highly non-commutative, while multiplication of real numbers is commutative, yet $\det(\cdot)$ turns one into the other
  • Miniquiz 15
• T:
  • Read: §§4.1 and 4.3
  • Content:
    1. properties of determinants:
       • determinants for triangular matrices (induction)
       • $\forall A,B,\ \det(AB)=\det(A)\cdot\det(B)$ (use elementary matrices)
       • $\det(A)=0\ \Leftrightarrow\ A$ is invertible. (elementary matrices, again)
    2. defining eigenvectors and eigenvalues
    3. examples of $2\times 2$ matrices with 0, 1, and 2 distinct eigenvalues
    4. Theorem: If $\lambda$ is an eigenvalue of the matrix $A$, then $\det(A-\lambda I_{n\times n})=0$.
  • Journal 5 on §§3.6 & 4.2 is due today
  • Hand in HW6: 4.2.53, 4.2.54, 4.2.69 (or it may be handed in tomorrow)
  • Miniquiz 16
• W:
  • Hand in HW6 if you did not do so yesterday
  • [Re]Read: §4.3 and Read: §4.4
  • Content:
    1. defining eigenspaces
    2. an eigenspace is always a subspace of $\RR^n$; in fact, it is the nullspace of $A-\lambda I$.
    3. the characteristic polynomial/equation of an $n\times n$ matrix
    4. the algebraic multiplicity of an eigenvalue
    5. the geometric multiplicity of an eigenvalue
    6. examples of eigenspaces and multiplicities of eigenvalues
  • Miniquiz 17
• F:
  • [Re]Read: §4.4
  • Content:
    1. definition of matrix similarity
    2. properties of matrix similarity:
       • reflexive
       • symmetric
       • transitive
       • ...so it's an equivalence relation
    3. properties in common to similar matrices
    4. diagonalizable matrices
  • Maxiquiz 6 was handed out today. Please work on it entirely on your own (although you may consult your book and notes) and hand it in on Tuesday as part of Midterm I. If you missed class, contact me and I will get you a copy of this maxiquiz for you to work on at home and then to hand in on Tuesday.
  • Hand in HW7: 4.1.35, 4.1.37, 4.3.20, and 4.3.21

• M:
  • [Re]Read: §4.4
  • Content:
    1. going over HW7
    2. review for Midterm I; see this review sheet
    3. eigenvalues of triangular (and hence, trivially, diagonal) matrices
    4. if $A\sim B$ then the eigenvalues of $A$ and $B$ are the same
    5. a $2\times 2$ matrix $A=\begin{pmatrix}a&b\\ 0&d\end{pmatrix}$ with $b\neq 0$ can be diagonalized — is similar to a matrix with the same diagonal entries but where the upper-right entry has been made $0$ — if $a\neq d$.
    6. $\forall A,\ \det(A)=\det(A^T)$ (just compute)
  • Miniquiz 18
• T:
  • Midterm I in class today
  • Hand in Maxiquiz 6, it will count as part of the Midterm.
  • No Journal is due today, although it might be a good idea to do one on the material in §§4.1, 4.3, and (at least part of) 4.4 as part of your studying for the midterm; the next journal will be due next Tuesday.
• W:
  • Content:
    1. going over Midterm I
• F:
  • [Re]Read: §4.4
  • Content:
    1. linear independence of eigenvectors corresponding to distinct eigenvalues
    2. building a basis for the ambient $\RR^n$ out of bases of all eigenspaces of an $n\times n$ matrix
       • a $2\times 2$ example
    3. The Diagonalization Theorem
    4. things to notice about an invertible matrix, say called $P$:
       • the columns of $P$ are a basis of $\RR^n$, call them $\vec{p}_1,\dots,\vec{p}_n$
       • conversely, if $\{\vec{p}_1,\dots,\vec{p}_n\}$ is a basis of $\RR^n$, and $P$ is a matrix whose columns are these vectors $\vec{p}_j$, for $1\le j\le n$, then $P$ is invertible $n\times n$.
       • multiplication on the left by $P$ transforms the standard basis of $\RR^n$ to the new basis consisting of the columns of $P$; that is, if $\vec{e}_1,\dots,\vec{e}_n$ is the standard basis (so $\vec{e}_j$ is really just the $j$th column of the $n\times n$ identity matrix, for $1\le j\le n$), then $\vec{p}_j=P\vec{e}_j,$ again for $1\le j\le n$.
       • multiplication on the left by $P^{-1}$ transforms the basis $\{\vec{p}_1,\dots,\vec{p}_n\}$ into the standard basis: $\vec{e}_j=P\vec{p}_j,$ for $1\le j\le n$.
    5. what this has to do with diagonalization:
       • if we can put together an entire basis $\{\vec{p}_1,\dots,\vec{p}_n\}$ of $\RR^n$ out of eigenvectors of some matrix $A$, so $A\vec{p}_j=\lambda_j\vec{p}_j$ for $1\le j\le n$, then building the matrix $P$ with columns from this basis, it will turn out that $P^{-1}AP$ is diagonal, with $\lambda_1,\dots,\lambda_n$ down the diagonal. [Why? because $$\left(P^{-1}AP\right)\vec{e_j}=P^{-1}\left(A\left(P\vec{e_j}\right)\right)=P^{-1}\left(A\vec{p_j}\right)=P^{-1}\left(\lambda_j\vec{p_j}\right)=\lambda_j\left(P^{-1}\vec{p_j}\right)=\lambda_j\vec{e}_j$$ so the standard basis of $\RR^n$ consists of eigenvectors of $P^{-1}AP$ with eigenvalues $\lambda_1,\dots,\lambda_n$ ... which means $P^{-1}AP$ is diagonal as claimed.]
    6. so we get the Diagonalization Theorem, two versions:
       • if separate bases of all the eigenspaces of an $n\times n$ matrix $A$ when put together yield a basis of all of $\RR^n$, then $A$ is diagonalizable.
       • if the geometric multiplicity equals the algebraic multiplicity for every eigenvalue of a matrix $A$, then $A$ is diagonalizable
    7. a corollary of the Diagonalization Theorem is that if an $n\times n$ matrix $A$ has $n$ distinct eigenvalues, then it is diagonalizable
    8. did examples of the practical process of diagonalizing a matrix we are given, following the above procedure
  • Hand in revised solutions to Midterm I, if you like; due to some students' scheduling conflict, you may also hand in revisions on Monday, if you prefer

• M:
  • Read: §5.1
  • Content:
    1. eigenvalues of invertible matrices
    2. orthogonal and linearly independent sets of vectors in $\RR^n$
    3. an orthonormal basis [ONB]
    4. coordinates with respect to an ONB — computing the coefficients by dot products
    5. defining orthogonal matrix and the set $O(n)$ of $n\times n$ orthogonal matrices
  • Miniquiz 19
  • last day to hand in revised solutions to Midterm I
  • Hand in HW8: one problem (your choice) in the set 4.4.{40, 41, 42} and 4.4.47
• T:
  • [Re]Read: §5.1
  • Content:
    1. 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 $O(n)$ of orthogonal matrices the orthogonal group.]
       • determinants of orthogonal matrices
  • Journal 6 on §§4.1, 4.3, & 4.4 is due today
  • Miniquiz 20
• W:
  • Read: §5.2
  • Hand in HW9: 5.1.26, 5.1.28(a)&(b) [read but do not do 5.1.28(c)&(d)]
  • Content:
    1. proved that give vectors $\vec{v}_1,\dots,\vec{v_k},\vec{w}_1,\dots,\vec{w}_\ell\in\RR^n$ such that $\vec{v}_i\perp\vec{w}_j$ $\forall i=1,\dots,k$, $\forall j=1,\dots,\ell$ (i.e., all the $\vec{v}$'s are orthogonal to all the $\vec{w}$'s) then $\Span(\vec{v}_1,\dots,\vec{v_k})\perp\Span(\vec{w}_1,\dots,\vec{w}_\ell)$
    2. defining orthogonal complement $W^\perp$ (pronounced "W-perp") of a subspace $W\subseteq\RR^n$
    3. proved that for any subspace of $\RR^n$, $W^\perp$ is a also a subspace of $\RR^n$
    4. for a matrix $A$: perps of column- and row-spaces, nullspaces and nullspaces of $A^T$ (stated without proofs)
  • Miniquiz 21
• F:
  • [Re]Read: §5.2
  • Content:
    1. orthogonal projections
    2. the Orthogonal Decomposition Theorem
  • Maxiquiz 7 today
  • Note: HW10, which was previously to be due today, has been delayed and changed: it will be due next Monday, and consists of a (slightly) different set of problems

• M:
  • Content:
    1. some discussion of Maxiquiz 7 and HW9
  • Hand in HW10: 5.2.25, 5.2.27
  • no miniquiz today, due to armadillos in the campus transformers
• T:
  • Read: §5.3 pp. 399-403 only
  • Content:
    1. going over Maxiquiz 7 and HW9
    2. brief revisit of orthogonal decomposition and projection
    3. the Gramm-Schmidt Process
  • Journal 7 on §§5.1, 5.2, & 5.3 (only pp. 399-403 in §5.3) is due today
  • OK to hand in HW10 today, due to yesterday's armadillos
  • Miniquiz 22
• W:
  • Read: §5.4
  • Content:
    1. defining orthogonally diagonalizable
    2. useful notation: "$\exists!x$..." means "there exists a unique $x$ ..."
    3. starting the Spectral Theorem: orthogonally diagonalizable implies symmetric
    4. example of orthogonally diagonalizing a symmetric matrix: key seems to be to find an ONB of $\RR^n$ consisting of eigenvectors
    5. mentioned (proof next time) that symmetric implies eigenspaces are orthogonal — this is the key step in the Spectral Theorem
  • OK to hand in Journal 7 today, due to Monday's armadillos
  • Miniquiz 23
• F:
  • [Re]Read: §5.4
  • Content:
    1. even more on the Spectral Theorem
    2. consequences of symmetry... distinct eigenspaces are orthogonal
    3. example application of the Spectral Theorem: the sum of two orthogonally diagonalizable matrices is also orthogonally diagonalizable
  • Maxiquiz 8 today
  • Hand in HW11: 5.3.3, 5.4.12, 5.4.14
  • Today [Friday] is the last day to withdraw (with a W) from classes

• M:
  • Read: §5.5 pp.425-432 (the section "Quadratic Forms") only
  • Content:
    1. yet more on the Spectral Theorem
    2. defining quadratic form
    3. examples of quadratic forms: upward- and downward-pointing paraboloids and saddles
    4. diagonalization of quadratic forms
    5. brick-throwing demonstration: which is related to the inertia tensor, and applying the Spectral Theorem yields something called The Principal Axes Theorem in physics
  • Miniquiz 24
• T:
  • Read: §6.1
  • Content:
    1. going over Maxiquiz 8 and HW11
    2. defining [abstract] vector space
    3. starting examples (and non-examples) of vector spaces:
       • $\RR^n$ with the usual vector addition and scalar multiplication
       • $\RR^2$ with modified vector addition(s) is often not a vector space
  • Miniquiz 25
  • Hand in HW12: 5.4.16, 5.5.38, 5.5.54
  • Journal 8 on §§5.4 & 5.5 (only pp. 425-432 "Quadratic Forms" in §5.5) is due today
• W:
  • [Re]Read: §6.1
  • Content:
    1. some algebraic (arithmetic?) properties in vector spaces which are consequences of their definition, e.g.:
       • in any vector space $V$, $0\vec{u}=\vec{0}\ \forall u\in V$
       • in any vector space $V$, $(-1)\vec{u}=-\vec{u}\ \forall u\in V$
       • in any vector space $V$, $\alpha\vec{0}=\vec{0}\ \forall \alpha\in\RR$
    2. more examples of vector spaces:
       • the trivial vector space $\{\vec{0}\}$
       • spaces of functions, such as:
         • $\Ff(\RR)$ — the space of all functions on the real line $\RR$
         • $C(\RR)$ — the space of continuous functions on the real line $\RR$
         • $C^k(\RR)$ for $k\in\NN$ — the space of $k$ times continuously differentiable functions on the real line $\RR$
         • $C^\infty(\RR)$ — the space of infinitely differentiable functions on the real line $\RR$
         • $\Pp_k$ for $k\in\NN$ — the space of polynomials in one variable of degree at most $k$
         • $\Pp$ — the space of all polynomials in one variable
         all with the pointwise addition of functions and scalar multiplications on functions
       • $M_{m\times n}$ — the space of $m\times n$ matrices, with the usual addition of matrices and scalar multiplication on matrices.
  • Miniquiz 26
• F:
  • [Re]Read: §6.1
  • Content:
    1. more discussion of the vector spaces of functions we defined last class -- these form a chain of subspaces:
       $\qquad\Pp_1\subseteq\Pp_2\subseteq\dots\subseteq\Pp\subseteq C^\infty(\RR)\subseteq\dots C^2(\RR)\subseteq C^1(\RR)\subseteq C(\RR)\subseteq\Ff(\RR)$
    2. defining [vector] subspace
    3. examples of subspaces
    4. how to check if something is a subspace (it's closed under vector addition and scalar multiplication
    5. defining Span in an abstract vector space
  • Maxiquiz 9 today

• M:
  • [Re]Read: §6.1 and Read: §6.2
  • Content:
    1. going over Maxiquiz 9 and recent HWs
    2. still more about vector subspaces, e.g., the $\vec{0}$ in a subspace is the same vector as the $\vec{0}$ in the ambient space
    3. more about Spans, e.g., as an intersection
    4. the subspaces of symmetric or skew-symmetric matrices in $M_{n\times n}$
    5. Cartesian product of vector spaces
    6. defining linearly [in]dependent in an abstract vector space, and examples
  • Hand in HW13: 6.1.{2, 6, 48}
  • Miniquiz 27
• T:
  • [Re]Read: §6.2
  • Content:
    1. defining basis in an abstract vector space, and examples
    2. defining dimension in an abstract vector space, and examples
    3. defining [in]finite dimensional for an abstract vector space, and examples
    4. the Basis Theorem in an abstract vector space
  • Journal 9 on §§6.1 & 6.2 (only the beginning of §6.2, if you like) is due today
  • Miniquiz 28
• W:
  • [Re]Read: §6.2 and Read: §6.3
  • Content:
    1. a little more care with finite and infinite sets of vectors which might be linearly [in]dependent
    2. coordinates of a vector $\vec{v}$ with respect to a basis $\Bb$ in an abstract vector space: $[\vec{v}]_\Bb\in\RR^k$, if $\Bb$ consists of $k$ basis vectors
    3. a change-of-basis matrix $P_{\Cc\leftarrow\Bb}$ for converting from basis $\Bb$ to basis $\Cc$, by $[\vec{v}]_\Cc=P_{\Cc\leftarrow\Bb}\,[\vec{v}]_\Bb$
    4. this $P_{\Cc\leftarrow\Bb}$ will be $k\times k$ if $\Bb$ (and thus also $\Cc$) consists of $k$ vectors
  • Miniquiz 29
• F:
  • [Re]Read: §6.3 and Read: §6.4
  • Content:
    1. more about coordinates, e.g., $P_{\Cc\leftarrow\Bb}$ is invertible: in fact $P_{\Cc\leftarrow\Bb}^{-1}=P_{\Bb\leftarrow\Cc}$
    2. definition of a linear tranformation between abstract vector spaces
  • Hand in HW14: 6.1.46, 6.2.34, 6.2.44
  • Maxiquiz 10 today

• M:
  • [Re]Read: §6.4
  • Content:
    1. going over Maxiquiz 10 and recent HWs
    2. examples of linear tranformations:
       • the zero transformation
       • the identity transformation
       • matrix multiplication
       • differentiation in $\Pp$
    3. properties of linear tranformations:
       • they map $\vec{0}_V$ to $\vec{0}_W$
       • they behave nicely with respect to the "additive inverse" operation $\vec{v}\mapsto-\vec{v}$.
       • compositions of LTs are LTs
• T:
  • [Re]Read: §6.4
  • Content:
    1. linear transformations are determined by what they do to a basis, which is, however, completely free: that is, if we want to make a linear transformation $T:V\to W$, and if $\{\vec{v}_1,\dots,\vec{v}_n\}$ is a basis of $V$, we can choose any vectors $\vec{w}_1,\dots,\vec{w}_n$ we like in $W$, and there will be a unique linear $T$ which satisifies $$T(\vec{v}_1)=\vec{w}_1,\quad\dots,\quad T(\vec{v}_n)=\vec{w}_n\ \ .$$
    2. more examples of linear tranformation
    3. inverses of linear transformations — are the linear?
  • Journal 10 on §§6.2 — 6.4 (the end of §6.2, if you did not finish it in Journal 9) is due today
  • Miniquiz 30
• W:
  • Content:
    1. review for Midterm II; see this review sheet
  • Hand in HW15: 6.3.16, 6.4.{20, 22, 24}
  • Miniquiz 31
• F:
  • Midterm II in class today

• M:
  • Content:
    1. going over Midterm II
• T:
  • Read: §6.5
  • Content:
    1. the kernel of a linear transformation
    2. the range of a linear transformation
    3. one-to-one (or 1-1 or injective)
    4. onto (or surjective)
    5. inverses of linear transformations
  • Hand in revised solutions to Midterm II, if you like
  • NOTE: no Journal is due today.
• W:
  • [Re]Read: §6.5
  • Content:
    1. kernels and ranges are always vector subspaces
    2. rank and nullity (again)
    3. The Rank-Nullity Theorem (again)
    4. isomorphisms and isomorphic
    5. a theorem giving a necessary and sufficient condition for finite dimensional vector spaces to be isomorphic
  • Miniquiz 32
• F:
  • Read: §6.6
  • Content:
    1. defining the matrix of a linear transformation with respect to bases of its domain and codomain
    2. the matrix of a composition of linear transformations
    3. the matrix of the inverse of a linear transformations
    4. matrices of endomorphisms of vector spaces and similarity using the change of basis matrix
  • Hand in HW16: 6.4.32, 6.5.27, & 6.5.34
  • Maxiquiz 11 today

• M:
  • Read: §7.1
  • Content:
    1. going over Maxiquiz 11
    2. an inner product and inner product space
    3. examples of inner product spaces:
       • $\RR^n$ with the usual dot product
       • the $L^2$ inner product on $C[0,1]$
       • $L^2$ also works on $\Pp$ and $\Pp_n$ ($n\in\NN$)
       • another inner product on $\Pp_n$: $\lt p,q\gt=p(0)q(0)+\dots+p(n)q(n)$ (just using the first term $p(0)q(0)$ works in all parts of the definition of an inner product except for non-degeneracy; with all $n+1$ terms it becomes a full inner product ... which we know because of the Fundamental Theorem of Algebra!)
  • Journal 11 on §§6.5 & 6.6 is due today
• T:
  • [Re]Read: §7.1
  • Content:
    1. elementary properties of inner products
    2. length or norm of vectors in an inner product space
    3. the Pythagorean Theorem
    4. orthogonal vectors in an inner product space
    5. projections and the Gram-Schmidt Process in an inner product space
    6. an orthonormal set in the inner product space $C[-\pi,\pi]$ with the $L^2$ inner product: trigonometric functions, and the connection with Fourier Analysis and radios
  • Miniquiz 33
• W:
  • Read: §7.2 pp.575-578 only
  • Content:
    1. the distance between vectors in an inner product space
    2. a norm and a normed linear space
    3. the distance function $d(\vec{u},\vec{v})$ in a normed linear space
    4. a metric and metric space
    5. examples of norms/metrics:
       • the norm coming from an inner product
       • the sum norm (also called the $L^1$ norm)
       • the max norm (also called the sup or $L^\infty$ norm)
       • the taxicab metric
  • Hand in HW17: 7.1.{36, 40}, 7.2.{8, 14}
  • Journal 12 on §§7.1 & 7.2 [only the parts we covered in this class] is due today
  • Maxiquiz 12 today
  • Hand in all late work and re-dos by 4pm today if you want them corrected and returned to you on Friday
• F:
  • Content:
    1. going over recent HWs
    2. review for the Final Exam next week; see this review sheet
  • Last day to hand in [by noon!] all late work and re-dos for class credit [although materials handed in today will not be returned]

• Exam week, no classes.
• Our [comprehensive] FINAL EXAM is scheduled for
  • Tuesday, December 10th, 10:30am-12:50pm in our usual classroom
  • Wednesday, December 11th, 10:30am-12:50pm in our usual classroom
  We will use both time slots. The format of the test is described on the final exam review sheet, but note that the parts in which you will have to state definitions and theorems will be on Tuesday (along with some other questions) — on Wednesday's part of the test, you will be able to refer to these definitions in your new work. So, in short: Wednesday is entirely problem-oriented, with little need for memorization, while for Tuesday, you will be required to have detailed, precise definitions and statements in your head.