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september 2

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\)

Fields, vector spaces, subspaces, linear operators, range space, null space

A field \((F,+,·)\) is a set with two commutative associative operations with \(0\) and\(1\), additive and multiplicative inverses and subject to the distributive law.

Examples (Common fields)

\((\Real,+,·)\) is a field. Rational, Complex numbers. Matrices (when they are fields?)

Vector spaces – modules over a field. (Real definition: abelian group with an action of \(\Field\) and several natural axioms.)

Examples

Fun(\(A,\Field\)), for any \(A\) – vector space over \(\Field\).…

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class of 8.29

    • express \(\sin(3x)\) in term of \(\sin(x)\).
    • express \(\cos(4x)\) in terms of \(\cos(x)\).
    • simplify
      \[
      \sum_{k=-20}^{k=20} e^{k\phi}.
      \]
  1. solve
    • \[
      y’=\frac{2t}{1+t^2}, y(0)=0;
      \]
    • \[
      y’=\frac{2t}{1-t^2}, y(0)=0;
      \]
    • \[
      y’=\frac{2t}{1-t^2}, y(2)=0.
      \]
  2. solve
    • \[
      y’=\frac{2x^2-4x+3}{x-1}, y(0)=2;
      \]
    • \[
      y’=\frac{x^2-4x+3}{x-2}, y(0)=2;
      \]
    • \[
      y’=\frac{x-2}{x^2-4x+3}, y(0)=2;
      \]
  3. Find with precision 5%
    \[
    \int_{-\pi}^\pi\cos(x)^{100} dx.
    \]

    (hint a: \(\cos(x)=(e^{ix}+e^{-ix})/2\); hint b: \(\cos(x)\approx 1-x^2/2\) for small \(x\)).

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august 28

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\)

What this course deals with?

Linear control systems. A continuous-time state-space linear system is defined by the following equations:
The signals
\[
x'(t) = A(t)x(t) + B(t)u(t), y(t) = C(t)x(t) + D(t)u(t),
\]
where
\[
x\in \Real^n, u\in\Real^k, y\in\Real^m;\\
y:[0,\infty)\to\Real^m,
u :[0,\infty)\to\Real^k,
x :[0,\infty)\to\Real^n,
\]
are the (vector-valued) functions called input, state, and output of the system.

The first-order differential equations is called the state and the output equation.

When all the matrices \(A(t), B(t), C(t), D(t)\) are constant, the system
is called a linear time-invariant (LTI) system.…

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ece 515, the week of August 25

No class August 26!

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math 285, the week of August 25

Brash up your calculus. Review Chapter 1.1. Listen to the first lecture.

Homework (due by Monday, 9.1):

  1. For which \(x\) there exists a real solution to
    \[
    y^2-4xy+4=0?
    \]
  2. Simplify
    \[
    \frac{\sin(x+y)-\sin(x-y)}{\sin(y)}.
    \]
  3. Solve differential equations
    \[
    y’=\sin(2x+3)
    \]
    and
  4. \[
    y’=\frac{1}{1+t^2}
    \]
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hyperbolic geometry of Google maps

\(\def\Real{\mathbb{R}}
\def\views{\mathbb{G}}
\def\earth{\mathbb{E}}
\def\hsp{\mathbb{H}}
\)

Google Maps and the Space of Views

Let’s consider the geometry hidden behind one of the best user interfaces in mobile apps ever, – the smartphone maps, the ones which you can swipe, flick and pinch. We are not talking here about the incredible engine that generates the maps on the fly as your finger waddles over the screen – just about the interface, which in a very intuitive and remarkably efficient way allows you to move between various maps.…

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