Date
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What we discussed/How we spent our time
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Aug 22
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Syllabus. Text.
Section 1.1:
Systems of linear equations. Augmented matrix of a linear system.
Elementary row operations.
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Aug 24
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Section 1.2:
Row reduction. (Reduced) row echelon form.
Pivots, pivot positions and pivot columns.
Free and pivot variables. Parametric form of
solution set.
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Aug 26
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We reviewed prior reading and lectures by working on
these questions. Then we defined
addition and multiplication of matrices and identified
some laws these operations satisfy. ($M_{m\times n}(\mathbb R)$
is a commutative group under $+, -, 0$.
Matrix multiplication is associative and
distributes over addition when the matrices have the appropriate
dimensions.) We saw that matrix multiplication is not commutative.
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Aug 29
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We discussed how to visualize vector arithmetic
in $\mathbb R^n$. We defined linear combination
and span. We discussed the possibilities for
$\textrm{Span}\; X$ when $X$ is a set of 0, 1, 2 or 3 vectors.
Quiz 1.
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Aug 31
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We reviewed the previous lecture and worked out an exercise
of the form: Determine if ${\bf b}$ belongs
to $\textrm{Span} \{{\bf v}_1,\ldots,{\bf v}_k\}$.
This led to the realization that
${\bf b}\in\textrm{Span} \{{\bf v}_1,\ldots,{\bf v}_k\}$
if and only if $A{\bf x}={\bf b}$ is consistent when
$A = \left[{\bf v}_1\;\cdots\;{\bf v}_k\right]$.
We discussed the geometric interpretation of this.
Then we discussed $A{\bf x}={\bf b}$ as a mapping problem.
Finally we stated and explained a theorem characterizing those matrices
$A$ with the property that $A{\bf x}={\bf b}$ is consistent
for all ${\bf b}$.
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Sep 2
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We started by working on
practice problems.
Then we discussed the structure of the coimage of the mapping
${\bf x}\mapsto A{\bf x}$. This led to the concept of a homogeneous system.
Note: next Monday's quiz has been moved to Wednesday.
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Sep 7
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We discussed the
decomposition of a function.
Then we took this quiz.
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Sep 9
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We worked out the answers to
this worksheet.
Then we discussed the general solution of a
homogeneous/nonhomogeneous linear system.
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Sep 12
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We discussed (trivial versus nontrivial) dependence relations,
and linear dependence/independence. We gave an algorithm for
determining if a set $Y$ of vectors is linearly independent.
Quiz.
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Sep 14
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We defined linear
transformation, and showed that matrix mappings
$T\colon \mathbb R^n\to \mathbb R^m\colon {\bf v}\mapsto A{\bf v}$
are linear.
We gave an example of a nonlinear transformation.
We defined the standard basis
$({\bf e}_1,\ldots,{\bf e}_n)$, and explained how to
find the standard matrix for a linear transformation.
We found the standard matrix for `differentiation of quadratic polynomials',
expressed as the transformation
$D\left(\left[\begin{array}{c}a\\b\\c\end{array}\right]\right) = \left[\begin{array}{c}2a\\b\end{array}\right]$.
I announced that a guest, Professor Agnes Szendrei, will lecture next week.
Everything will go as usual (e.g., quiz on Monday, HW due Wednesday).
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Sep 16
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We worked out the answers to
this worksheet.
Then we discussed how to find matrices for
rotations with respect to the origin,
reflections through a line through the origin,
and dilations with respect to the origin
in $2$-dimensional space.
I announced that a guest, Professor Agnes Szendrei, will lecture next week.
Everything will go as usual (e.g., quiz on Monday, HW due Wednesday).
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Sep 19
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Applications: network flow and nutritional diet.
Quiz.
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Sep 21
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Handout.
Inverse of a matrix: definition, basic properties and
algorithm for finding the inverse.
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Sep 23
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Various characterizations of invertible matrices.
Inverse of a linear transformation.
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Sep 26
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Review of left, right and 2-sided invertibility.
We started discussing subspaces, bases and dimension.
Quiz.
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Sep 28
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We discussed the column space algorithm and the null space algorithm.
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Sep 30
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We discussed homogeneous coordinates, affine tranformations,
said a few words about partitioned matrices, and
explained how to find the $3\times 3$-matrix
representing (in homogeneous coordinates)
a given affine rotation of the plane.
Practice problems!.
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Oct 3
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We worked out an example showing how to find the matrix that reflects
a $2$-dimensional vector in homogeneous coordinates through
the line of slope $-1$ that passes through the point $(1,1)$.
We next began to discuss length, area and volume.
We derived the area formula
$A\left(
\left[\begin{array}{c}a\\c\end{array}\right],
\left[\begin{array}{c}b\\d\end{array}\right]\right)=ad-bc$.
Quiz.
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Oct 5
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We worked on practice problems.
As we discussed the solutions, we observed that
any two bases of a subspace have the same size.
We then discussed length, area, and volume in
$\mathbb R^1$, $\mathbb R^2$, and $\mathbb R^n$, $n\geq 3$.
We evolved a definition of signed volume in $\mathbb R^n$:
a multilinear alternating function normalized to $1$
on the unit hypercube.
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Oct 7
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We discussed the
determinant.
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Oct 10
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We continued discussing the determinant.
I circulated a
review sheet.
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Oct 12
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We reviewed for the midterm. I circulated the
quiz
that I forgot to
give on Monday.
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Oct 14
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Midterm.
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Oct 17
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Cramer's rule. The Vandermonde determinant.
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Oct 19
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Abstract vector spaces: definition and examples.
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Oct 21
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We worked on practice problems. We briefly discussed
last week's
midterm. Then we proved that if $S$ is a finite spanning set for
a vector space
$\mathbb V$, then $S$ contains a basis for $\mathbb V$. We defined the
$\mathcal B$-coordinate vector, $\left[{\mathbf u}\right]_{\mathcal B}$,
for a vector ${\mathbf u}\in\mathbb V$ with respect to an
ordered basis $\mathcal B$.
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Oct 24
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We reviewed the meaning of isomorphism (= invertible
linear transformation whose inverse is also linear).
We explained why
a linear transformation that is 1-1 and onto is an isomorphism.
We proved that a finitely generated real vector space is isomorphic
to $\mathbb R^n$ for some $n$.
Quiz.
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Oct 26
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We discussed matrices for linear transformations,
in particular change of basis matrices
${}_{\mathcal C}[I]_{\mathcal B}$.
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Oct 28
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Practice problems about coordinates.
We discussed why $[{\bf v}]_{\mathcal B}$ may be calculated
by solving $[{\mathcal B}][{\bf x}] = [{\bf v}]$, or may
be computed as
$[{\mathcal B}]^{-1}[{\bf v}]$, and why ${}_{\mathcal C}[I]_{\mathcal B}$
may be computed as $[{\mathcal C}]^{-1}[{\mathcal B}]$.
We defined $\mathbb V+\mathbb W$ and $\mathbb V\cap\mathbb W$,
and explained how to compute bases for these subspaces
given bases for $\mathbb V$ and $\mathbb W$.
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Oct 31
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We worked on practice problems
and took a
quiz.
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Nov 2
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We discussed e-values and e-vectors. (Read 265-276.)
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Nov 4
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We worked on practice problems,
then discussed how to compute a particular coefficient
of $\chi_A(\lambda) = \det(\lambda I-A)$. Namely, the coefficient
$s_i$ of $\lambda^{n-i}$ is $(-1)^i$ times the sum of the
$i\times i$ principal minors of $A$. The most important
special cases are $s_1=\textrm{tr}(A)$ and $s_n=\det(A)$.
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Nov 7
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We discussed the definitions of algebraic multiplicity
and geometric multiplicity of an e-value, and computed
those numbers in $2$ examples.
Quiz.
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Nov 9
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We discussed the direct sum
${\mathbb U}\oplus{\mathbb W}$ of two
subspaces of $\mathbb V$.
We proved that a set of e-vectors for distinct e-values
is independent. We derived that if $A\in M_{n\times n}(\mathbb R)$
has $n$ distinct e-values, then $\mathbb R^n$ is a direct sum of the
1-dimensional e-spaces of $A$. We proved that if $\mathcal B$
is a basis of e-vectors for $A$, then ${}_{\mathcal B}[A]_{\mathcal B}$
is a diagonal form for $A$ which has the e-values of $A$
on the diagonal.
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Nov 11
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We worked on practice problems.
Then discussed why the characteristic polynomial
of a matrix is unaffected
by a change of basis, and why the algebraic multiplicity of
an e-value is greater or equal to the geometric multiplicity
of the e-value.
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Nov 14
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We discussed complex numbers, complex e-values,
and complex e-vectors. We discussed diagonalization
over complex numbers and block diagonalization over the real
numbers.
Quiz.
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Nov 16
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We worked out Extra Problem 3 on the HW.
Then we discussed the Cayley-Hamilton Theorem,
defined minimal polynomial, and asserted that a
matrix $A$ is diagonalizable iff $\textrm{min}_A(t)$
factors into distinct linear factors.
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Nov 18
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We worked on practice problems
about linear transformations of order $2$. Then we discussed
an application of diagonalization to the solution of
ordinary differential equations. Namely, we showed how to reduce a single
$n$-th order, homogeneous, ODE with constant coefficients
to a system of $n$ first-order equations, then (when diagonalizable)
to a system of $n$ equations of the form $z'=\lambda z$.
We showed that an equation like this has solution $z=C e^{\lambda t}$.
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Nov 28
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We worked on practice problems
and took a quiz.
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Nov 30
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We discussed dot product, length, angle, orthogonality,
and orthogonal complements.
We explained why
${\bf u}\bullet{\bf v}=\|{\bf u}\|\cdot\|{\bf v}\|\cdot\cos(\theta)$.
We proved that if $X\subseteq \mathbb R^n$, then $X^{\perp}$ is
a subspace.
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Dec 2
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We described an algorithm to compute $X^{\perp}$ for any
subset $X\subseteq \mathbb R^n$.
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Dec 5
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We proved that if $U\leq \mathbb R^n$ is a subspace,
then $\mathbb R^n = U\oplus U^{\perp}$ and $U^{\perp\perp} = U$.
We introduced the normal equations, $A^TA{\bf x}=A^T{\bf b}$,
and worked on this worksheet.
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Dec 7
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We discussed the method of least squares (Section 6.5).
We worked on this worksheet.
I circulated this review sheet.
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Dec 9
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We reviewed for the final exam.
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