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.

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
  • 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:
  • [Re]Read: §3.3
  • Content:
    1. going over Maxiquiz 3 — 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., in class we discussed an inductive proof that all pigs are yellow [which is unfortunately incorrect ... extra credit points to whomever can find the flaw in the proof!] — so your best bet is to stick carefully to the standard structure of induction proofs and to check over each step very carefully
    2. discussion of where incautious induction can lead to false proofs (e.g., the proof that all pigs are yellow)
    3. elementary matrices
    4. the Fundamental Theorem of Invertible Matrices (version 1.0)
    5. discussion about how 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.
• T:
  • Read: §3.5
  • Content:
    1. discussed why the "all pigs are yellow" proof fails -- the inductive step doesn't work unless $n > 2$, so either that step needs a better proof, or one needs a different base case.
    2. the idea of a subspace of $\RR^n$
    3. examples of subspaces of $\RR^2$ (in fact, these are all possible subspaces):
       • $\{\vec{0}\}$, the trivial subspace
       • any line through the origin
       • all of $\RR^2$
    4. 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
    5. $\forall n,k\in\NN$ and $\forall\vec{v}_1,\dots,\vec{v}_k\in\RR^n$, $\Span(\vec{v}_1,\dots,\vec{v_k})$ is a subspace of $\RR^n$
    6. subspaces associated with a given matrix:
       • the row space
       • the column space
  • Journal 3 on §§3.5 & 3.6 is NOT due this week... but don't forget to start thinking about it, there is a lot of material in these two sections. It will be due next Tuesday.
  • Miniquiz 8
• W:
  • [Re]Read: §3.5
  • Content:
    1. note that the definition of a subspce of $\RR^n$ would be equivalent to the one we use even without the specific requirement that $\vec{0}$ is in the subset, as long as the other parts are retained
    2. basis, examples of bases
    3. dimension, computations of dimensions for simple cases
  • Miniquiz 9
• F:
  • [Re]Read: §3.5
  • Content:
    1. rank
    2. the Fundamental Theorem of Invertible Matrices 2.0
  • Hand in HW5: 3.5.4, 3.5.34
  • Maxiquiz 4 today

• M:
  • Read: §3.6
  • Content:
    1. going over Maxiquiz 4
    2. the null space of a matrix
    3. nullity
    4. The Rank[-Nullity] Theorem
  • Hand in HW6: 3.5.56, 3.5.60 (these are serious problems — don't hesitate to contact me and ask for a hint).
• T:
  • [Re]Read: §3.6
  • Content:
    1. The Basis Theorem
    2. coordinates w.r.t. a basis (what amounts to Theorem 3.29 in §3.5)
    3. a linear transformation
    4. examples of linear transformations:
       • the trivial transformation which sends every input to $\vec{0}$.
       • some random formulæ
       • rotations of $\RR^2$
       • reflections of $\RR^2$
    5. a Proposition: If $f:\RR^n\to\RR^M$ is a linear transformation, then $f(\vec{0})=\vec{0}$.
  • Journal 3 on §3.5 is due today
  • Miniquiz 10
• W:
  • [Re]Read: §3.6
  • Content:
    1. more examples of linear transformations
    2. a linear transformation coming from matrix multiplication — what the book (and almost no one else) calls a "matrix transformation"
    3. a linear transformation is determined by what it does to a basis
    4. the matrix $[T]$ of a linear transformation $T$, with examples:
       • a counterclockwise rotation of $\RR^2$ by the angle $\theta$ has matrix $\begin{pmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{pmatrix} $.
       • a reflection of $\RR^2$ across the $y$-axis has matrix $\begin{pmatrix}-1&0\\ 0&1\end{pmatrix}$ .
       • projection onto the $x$-axis in $\RR^2$ has matrix $\begin{pmatrix}1&0\\ 0&0\end{pmatrix}$ .
    5. composition of linear transformations
       • it's linear, too
       • its matrix is the product of the matrices of the consituent transformations
  • Miniquiz 11
• F:
  • Read: §4.2
  • Content:
    1. recall 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
  • Maxiquiz 5 today
  • Hand in HW7: 3.6.4, 3.6.8, 3.6.44

• M:
  • [Re]Read: §4.2 and Read: §4.1
  • Content:
    1. going over Maxiquiz 5
    2. the identity transformation of $\RR^n$
    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 $n\ge 3$ — which is essentially Laplace's Expansion Theorem
       • [there is actually a direct (=non-recursive) formula]
    5. properties of determinants:
       • $\det(A)=0\ \Leftrightarrow\ A$ is invertible.
       • $\forall A,\ \det(A)=\det(A^T)$
       • $\forall A,B,\ \det(AB)=\det(A)\cdot\det(B)$
       • determinants for triangular matrices
  • Miniquiz 12
• T:
  • [Re]Read: §4.1 and Read: §4.3
  • Content:
    1. eigenvectors, eigenvalues, and 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
  • Journal 4 on §§3.6 & 4.2 is due today
  • Hand in HW8: 4.2.53, 4.2.54, 4.2.69
  • Miniquiz 13
• W:
  • [Re]Read: §4.4 and Read: §4.4
  • Content:
    1. the geometric multiplicity of an eigenvalue
    2. definition of matrix similarity
    3. examples of eigenspaces and multiplicities of eigenvalues
  • Miniquiz 14
• F:
  • [Re]Read: §4.4
  • Content:
    1. eigenvalues of triangular matrices
    2. eigenvalues of invertible matrices
    3. linear independence of eigenvectors corresponding to distinct eigenvalues
    4. properties of matrix similarity:
       • reflexive
       • symmetric
       • transitive
       • ...so it's an equivalence relation
    5. properties in common to similar matrices
    6. diagonalizable matrices
    7. 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
    8. The Diagonalization Theorem
  • Maxiquiz 6 today
  • Hand in HW9: 4.1.35, 4.1.37, 4.3.20-22

• M:
  • Content:
    1. going over Maxiquiz 6 and HW9
    2. review for Midterm I; see this review sheet
    3. discuss any issues with 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
• F:
  • [Re]Read: §5.1
  • Content:
    1. 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 $1\le j\le n$.
    2. 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.]
    3. 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
    4. a corollary of the Diagonalization Theorem is that if an $n\times n$ matrix $A$ has $n$ distinct eigenvalues, then it is diagonalizable
  • Hand in revised solutions to Midterm I, if you like

• M:
  • [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 — computing the coefficients by dot products
    4. defining orthogonal matrix and the set $O(n)$ of $n\times n$ orthogonal matrices
  • Miniquiz 16 (on §5.1!)
• 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
  • Miniquiz 17
• W:
  • Read: §5.2
  • Hand in HW11: 5.1.26, 5.1.28(a)&(b) [read but do not do 5.1.28(c)&(d)]
  • Content:
    1. defining orthogonal complement $W^\perp$ (pronounced "W-perp") of a subspace $W\subseteq\RR^n$
    2. for a matrix $A$: perps of column- and row-spaces, nullspaces and nullspaces of $A^T$
  • Miniquiz 18
• F:
  • [Re]Read: §5.2 and Read: §5.3 pp. 385-389 only
  • Content:
    1. orthogonal projections
    2. the Orthogonal Decomposition Theorem
    3. the Gramm-Schmidt Process
  • Maxiquiz 7 handed out today
  • Hand in HW12: 5.2.27, 5.2.29

• M:
  • Read: §5.4
  • Content:
    1. some discussion of problems from HW12
    2. defining orthogonally diagonalizable
    3. starting the Spectral Theorem: orthogonally diagonalizable implies symmetric
  • Journal 6 on §§5.1, 5.2, & 5.3 (only pp. 385-389 in 5.3) is due today
  • Hand in Maxiquiz 7
  • Hand in HW13: 5.2.25, 5.3.3
  • Miniquiz 19
• T:
  • [Re]Read: §5.4
  • Content:
    1. consequences of symmetry... having real eigenvalues...
    2. more on the Spectral Theorem
  • Miniquiz 20
• W:
  • [Re]Read: §5.4
  • Content:
    1. even more on the Spectral Theorem — examples
  • Hand in HW14: 5.4.12, 5.4.14, 5.4.16
  • Miniquiz 21
• F:
  • Read: §5.5 pp.411-426 only
  • Content:
    1. yet more on the Spectral Theorem
    2. brick-throwing demonstration
  • Maxiquiz 8 today
  • Today [Friday] is the last day to withdraw (with a W) from classes

• M:
  • [Re]Read: §5.5 pp.411-426 only
  • Content:
    1. defining quadratic form
    2. examples of quadratic forms: upward- and downward-pointing paraboloids and saddles
    3. diagonalization of quadratic forms
    4. our throwing bricks demo from last week is related to the inertia tensor, and applying the Spectral Theorem yields something called The Principal Axes Theorem in physics
• T:
  • Read: §6.1
  • Content:
    1. defining vector space
    2. starting examples (and non-examples) of vector spaces:
       • the trivial vector space $\{\vec{0}\}$
       • $\RR^n$ with the usual vector addition and scalar multiplication
       • $\RR^2$ with modified vector addition(s) is often not a vector space
  • Start HW15, which is due tomorrow
  • Miniquiz 22
• 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 $V$, $0\vec{u}=\vec{0}\ \forall u\in V$).
    2. more examples of vector spaces:
       • 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.
  • Hand in HW15: 5.5.32, 5.5.38, 5.5.42
  • Miniquiz 23
• 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
  • 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
    3. another vector space example: $M_{m\times n}$ — the space of $m\times n$ matrices, with the usual addition of matrices and scalar multiplication on matrices.
    4. the subspaces of symmetric or skew-symmetric matrices in $M_{n\times n}$
  • Hand in HW16: 6.1.{2, 6, 12, 48}
  • Miniquiz 24
• T:
  • [Re]Read: §6.2
  • Content:
    1. defining Span in an abstract vector space
    2. properties of Span in an abstract vector space
    3. defining linearly [in]dependent in an abstract vector space, and examples
    4. defining basis in an abstract vector space, and examples
    5. defining dimension in an abstract vector space, and examples
  • Journal 7 on §§5.4, 5.5 (only pp. 411-418), & 6.1 is due today
  • Miniquiz 25
• W:
  • [Re]Read: §6.2
  • Content:
    1. a little more care with finite and infinite sets of vectors which might be linearly [in]dependent
    2. defining [in]finite dimensional for an abstract vector space, and examples
    3. the Basis Theorem in an abstract vector space
  • Miniquiz 26
• F:
  • [Re]Read: §6.2
  • Content:
    1. yet more care with finite and infinite sets of vectors which might be linearly [in]dependent
    2. be careful of the definition of dimension in the book, it doesn't quite work!
    3. 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
  • Hand in HW17: 6.1.46, 6.2.{6, 34, 44}
  • Maxiquiz 10 today

• M:
  • Read: §6.3 and [Re]Read: §6.4
  • Content:
    1. going over Maxiquiz 10 and recent HWs
    2. still more about coordinates:
       • a change-of-basis matrix $P_{\Cc\leftarrow\Bb}$ for converting from basis $\Bb$ to basis $\Cc$, by $[\vec{v}]_\Cc=P_{\Cc\leftarrow\Bb}\,[\vec{v}]_\Bb$
       • this $P_{\Cc\leftarrow\Bb}$ will be $k\times k$ if $\Bb$ (and thus also $\Cc$) consists of $k$ vectors
       • $P_{\Cc\leftarrow\Bb}$ is invertible: in fact, $P_{\Cc\leftarrow\Bb}^{-1}=P_{\Bb\leftarrow\Cc}$
  • Miniquiz 27
• T:
  • Read: §6.4
  • Content:
    1. more examples of change-of-basis matrices
    2. definition of a linear tranformation between abstract vector spaces
  • Miniquiz 28
• W:
  • [Re]Read: §6.4
  • Content:
    1. examples of linear tranformations
       • the zero transformation
       • the identity transformation
       • matrix multiplication
       • differentiation in $\Pp$
    2. review for Midterm II; see this review sheet
  • Hand in HW18: 6.3.16, 6.3.21 [Hint: use Thm 6.12b.], 6.4.{20, 22, 24}
  • Miniquiz 29 (handed out as a take-home quiz, due Friday)
• F:
  • Midterm II in class today
  • Journal 8 on §§6.2-6.4 is due today

• M:
  • Content:
    1. going over Midterm II
• 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. composition of linear transformations
    3. inverses of linear transformations
  • Hand in revised solutions to Midterm II, if you like
• W:
  • Read: §6.5
  • Content:
    1. last words about inverses of lienar transformations
    2. the kernel of a linear transformation
    3. the range of a linear transformation
    4. one-to-one (or 1-1 or injective)
    5. onto (or surjective)
  • Miniquiz 30
• F:
  • Read: §6.6
  • Content:
    1. kernels and ranges are always vector subspaces
    2. rank and nullity (again)
    3. The Rank-Nullity Theorem (again)
    4. isomorphisms and isomorphic
  • Hand in HW19: 6.4.{26, 32}, 6.5.{4, 24, 27, 34}
  • Maxiquiz 11 today

• M:
  • Read: §7.1
  • Content:
    1. going over Maxiquiz 11
    2. a theorem giving a necessary and sufficient condition for finite dimensional vector spaces to be isomorphic
    3. an inner product and inner product space
    4. examples of inner product spaces:
       • $\RR^n$ with the usual dot product
       • the $L^2$ inner product on $C[0,1]$
  • Journal 9 on §§6.5 & 6.6 is due today
  • Miniquiz 31
• 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 32
• W:
  • Read: §7.2 pp.561-564 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 HW20: 7.1.{36, 40}, 7.2.{8, 14}
  • Journal 10 on §§7.1 & 7.2 [only the parts we covered in this class] is due today
  • Miniquiz 33
  • 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
  • Monday, April 30th, 8:00-10:20 in our usual classroom
  • Tuesday, May 1st, 8:00-10:20 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 Monday (along with some other questions) — on Tuesday's part of the test, you will be able to refer to these definitions in your new work. So, in short: Tuesday is entirely problem-oriented, with little need for memorization, while for Monday, you will be required to have detailed, precise definitions and statements in your head.