Linear Algebra Homework 1

Linear Algebra - Homework 1

• Homework due on Thursday, January 22 at 17:00 in office 3355 of the André-Aisenstadt building.

• This homework counts for 5% of total grade.

• If $A \in M_{n \times n}(\mathbb{R})$ is a nilpotent matrix, then $A$ is similar to a matrix where all diagonal elements are zero.

• If $A, B \in M_{n \times n}(\mathbb{C})$ such that $AB = BA$ and $A$ is invertible, then there exists a polynomial $p(x) \in \mathbb{C}[x]$ such that $B = p(A)$.

• For any $A \in M_{n \times n}(\mathbb{C})$, $A$ and $A^t$ are similar matrices.

• Given the matrix $A = \begin{pmatrix} 5 & -6 & -6 \ -1 & 4 & 2 \ 3 & -6 & -4 \end{pmatrix}$, there exists an invertible matrix $P \in M_{3 \times 3}(\mathbb{R})$ such that $P^{-1}AP$ is in Jordan form.

• $V$ is the vector space of real polynomials with degree at most $n$, denoted as $\mathbb{R}_n[x]$.

• $T : V \to V$ is defined by $T(p(x)) = p(x) + p'(x) + p''(x)$, where $p'(x)$ and $p''(x)$ are the first and second derivatives of $p(x)$, respectively.

• $T$ is a linear operator.

• $T$ is invertible.

• The matrix of $T$ with respect to the base $\mathcal{B} = {1, x, x^2,..., x^n}$ can be determined.

• The matrix of $T^{-1}$ with respect to the base $\mathcal{B}$ can be determined.

Lab 1: Physical Layer

• Focuses on the physical layer of the OSI model, which transmits raw bits over a communication channel.
• Different tools will be used to analyze physical layer characteristics like bandwidth, noise, and attenuation.

Part 1: Signal Generation and Spectrum Analysis

• Use an arbitrary waveform generator to create sine wave signals and vary their frequency and amplitude.
• Observe the generated signals using an oscilloscope.
• Use a spectrum analyzer to observe the frequency domain representation of the generated signals.
• Spectrum analysis involves identifying the fundamental frequency and harmonics, and analyzing the signal's bandwidth.
• Implement Amplitude Modulation (AM) and Frequency Modulation (FM) using the waveform generator.
• The spectrum and bandwidth of modulated signals need to be analyzed.

Part 2: Channel Impairments

• Different types of noise (e.g., white noise, impulse noise) are introduced into the communication channel.
• Observe the effect of noise on the transmitted signals and measure the Signal-to-Noise Ratio (SNR).
• Introduce attenuation in the communication channel using attenuators or long cables.
• Signal strength needs to be measured at different points in the channel to analyze the impact of attenuation on signal quality.
• Use different lengths of cables to introduce delay distortion and observe its effect on the transmitted signals.

Part 3: Error Detection and Correction

• Implement a parity check scheme for error detection, simulate bit errors in the transmitted data, and evaluate its effectiveness.
• Implement a CRC algorithm for error detection and compare its performance with the parity check scheme.
• Implement a Hamming code for error detection and correction and evaluate its error correction capability.

Part 4: Channel Capacity

• Calculate the channel capacity using the Shannon-Hartley theorem: $C = B \log_2(1 + \frac{S}{N})$
  • $C$ is the channel capacity in bits per second (bps).
  • $B$ is the bandwidth of the channel in Hertz (Hz).
  • $S$ is the average signal power.
  • $N$ is the average noise power.
• Vary the bandwidth and SNR to analyze their impact on channel capacity.
• Calculate the maximum data rate using the Nyquist theorem: $Maximum Data Rate = 2 \cdot B \cdot \log_2(V)$
  • $B$ is the bandwidth of the channel in Hertz (Hz).
  • $V$ is the number of discrete signal levels.
• Analyze the impact of bandwidth and signal levels on the maximum data rate.

Deliverables

• A lab report including introduction, objectives, description of the experimental setup, and results/analysis for each part.
• The report should also include a conclusion and discussion of the key findings, along with code implementation of the error detection and correction schemes.

Complex Numbers

• A complex number is a number that can be expressed in the form $a + bi$.
  • $a$ and $b$ are real numbers.
  • $i$ is the imaginary unit, where $i^2 = -1$.

Components

• Real Part: $a$.
• Imaginary Part: $b$.

Examples:

• $3 + 2i$: Real part is 3, imaginary part is 2.
• $-1 - i$: Real part is -1, imaginary part is -1.
• $4i$: Real part is 0, imaginary part is 4.
• $5$: Real part is 5, imaginary part is 0.

Operations with Complex Numbers

• Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$. Example: $(3 + 2i) + (1 - i) = 4 + i$
• Subtraction: $(a + bi) - (c + di) = (a - c) + (b - d)i$. Example: $(5 - 3i) - (2 + i) = 3 - 4i$
• Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$. Example: $(2 + i)(3 - 2i) = 8 - i$
• Division: Multiply the numerator and denominator by the conjugate of the denominator. $\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$. Example: $\frac{1 + 2i}{3 - i} = \frac{1}{10} + \frac{7}{10}i$

Complex Conjugate

• The complex conjugate of $a + bi$ is $a - bi$, denoted as $\overline{a + bi} = a - bi$.

Properties of Complex Conjugates:

• $\overline{z + w} = \overline{z} + \overline{w}$.
• $\overline{z - w} = \overline{z} - \overline{w}$.
• $\overline{zw} = \overline{z} \cdot \overline{w}$.
• $\overline{(\frac{z}{w})} = \frac{\overline{z}}{\overline{w}}$.
• $\overline{\overline{z}} = z$.
• $z + \overline{z} =$ 2Re$(z)$, which is a real number.
• $z - \overline{z} =$ 2iIm$(z)$, which is an imaginary number.
  • Where $z$ and $w$ are complex numbers.

Modulus of a Complex Number

• The modulus (or absolute value) of $z = a + bi$ is $|z| = \sqrt{a^2 + b^2}$. It is the distance from the origin to the point $(a,b)$ in the complex plane.

Properties of Modulus:

• $|z| \geq 0$.
• $|z| = 0$ if and only if $z = 0$.
• $|zw| = |z| \cdot |w|$.
• $|\frac{z}{w}| = \frac{|z|}{|w|}$, if $w \neq 0$.
• $|z + w| \leq |z| + |w|$ (Triangle Inequality).
• $|z - w| \geq ||z| - |w||$.

Polar Form of Complex Numbers

• A complex number $z = a + bi$ can be represented in polar form as: $z = r(\cos\theta + i\sin\theta)$.
  • $r = |z| = \sqrt{a^2 + b^2}$ is the modulus of $z$.
  • $\theta$ is the argument of $z$, denoted as $\arg(z)$, and is the angle between the positive real axis and the line connecting the origin to the point $(a, b)$ in the complex plane.
• Finding $\theta$: $\theta = \arctan\left(\frac{b}{a}\right)$
• Euler's Formula: $e^{i\theta} = \cos\theta + i\sin\theta$

Formula:

• Using Euler's formula, the polar form of a complex number can be written as: $z = re^{i\theta}$.
• De Moivre's Theorem states that for any complex number in polar form $z = r(\cos\theta + i\sin\theta)$ and any integer $n$: $z^n = [r(\cos\theta + i\sin\theta)]^n = r^n(\cos(n\theta) + i\sin(n\theta))$.
• Using Euler's form: $(re^{i\theta})^n = r^n e^{in\theta}$.
• De Moivre's Theorem is useful for finding powers and roots of complex numbers.
• To find the n-th roots of a complex number $z = r(\cos\theta + i\sin\theta)$, use the formula: $w_k = \sqrt[n]{r}\left(\cos\left(\frac{\theta + 2\pi k}{n}\right) + i\sin\left(\frac{\theta + 2\pi k}{n}\right)\right)$ where $k = 0, 1, 2,..., n-1$.

The Economics of Money, Banking, and Financial Markets - Contents

• Book is the Twelfth Edition by Frederic S. Mishkin from Columbia University

Part I: Why Study Money, Banking, and Financial Markets?

• Chapter 1: Why Study Money, Banking, and Financial Markets?
• Chapter 2: An Overview of the Financial System

Part II: Central Banking and the Money Supply Process

• Chapter 3: What is Money?
• Chapter 4: Understanding Interest Rates
• Chapter 5: The Behavior of Interest Rates
• Chapter 6: The Risk and Term Structure of Interest Rates
• Chapter 7: The Stock Market, the Theory of Rational Expectations, and the Efficient Market Hypothesis
• Chapter 8: An Economic Analysis of Financial Structure
• Chapter 9: Financial Crises
• Chapter 10: Central Banks and the Federal Reserve System
• Chapter 11: The Money Supply Process
• Chapter 12: Tools of Monetary Policy
• Chapter 13: The Conduct of Monetary Policy: Strategy and Tactics

Part III: Financial Institutions

• Chapter 14: The Foreign Exchange Market
• Chapter 15: The International Financial System

Part IV: International Finance and Monetary Policy

• Chapter 16: Monetary Policy Strategy: Lessons from the Financial Crisis
• Chapter 17: Aggregate Demand, and Supply Analysis
• Chapter 18: Monetary Policy Theory
• Chapter 19: The Role of Expectations in Monetary Policy
• Chapter 20: Inflation Targeting
• Chapter 21: Financial Stability and Macroprudential Policy
• Chapter 22: The IS Curve
• Chapter 23: The Monetary Policy and Aggregate Demand Curves
• Chapter 24: Aggregate Supply
• Chapter 25: Monetary Policy Theory: A More Detailed Treatment
• Chapter 26: Transmission Mechanisms of Monetary Policy

Chemical Kinetics

Reaction Rate

• Reaction rate refers to a change in concentration of reactant or product with respect to time
• For a reaction $aA + bB \rightarrow cC + dD$, rate equation is: $Rate = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt}$.
  • $[A]$, $[B]$, $[C]$, $[D]$ are the concentrations of reactants and products.
  • $a$, $b$, $c$, $d$ are the stoichiometric coefficients.

Rate Law

• The rate law expresses the relationship between the rate of a reaction and the concentrations of the reactants.
• General form: $Rate = k[A]^m[B]^n$
  • $k$ is the rate constant.
  • $m$ and $n$ are the reaction orders with respect to reactants A and B, respectively.
  • $m + n$ is the overall reaction order.
• Reaction orders are experimentally determined.

Factors Affecting Reaction Rate

• Increasing temperature, concentration of reactants, surface area of solid reactants (for heterogeneous reactions), and pressure (for gaseous reactions) generally increases the reaction rate.
• Catalysts increase the reaction rate.

Rate Constant (k)

• The rate constant ($k$) is a proportionality constant that relates the rate of a reaction to the concentrations of reactants.
• Arrhenius Equation: $k = Ae^{-\frac{E_a}{RT}}$
  • $A$ is the pre-exponential factor (frequency factor).
  • $E_a$ is the activation energy.
  • $R$ is the ideal gas constant ($8.314 , J/(mol \cdot K)$).
  • $T$ is the absolute temperature in Kelvin.
• The activation energy is the minimum energy required for a reaction to occur.

Reaction Mechanisms

• A reaction mechanism is a step-by-step sequence of elementary reactions by which overall chemical change occurs.
• An elementary step is a single step in a reaction mechanism.
• The rate-determining step is the slowest step in a reaction mechanism and determines the overall rate of the reaction.

Catalysis

• Homogeneous: The catalyst is in the same phase as the reactants.
• Heterogeneous: The catalyst is in a different phase from the reactants.
• Enzymes: Biological catalysts that catalyze biochemical reactions.

Integrated Rate Laws

• Zero-Order Reactions
  • Rate Law: $Rate = k$
  • Integrated Rate Law: $[A]_t = -kt + [A]_0$
  • Half-Life: $t_{1/2} = \frac{[A]_0}{2k}$
• First-Order Reactions
  • Rate Law: $Rate = k[A]$
  • Integrated Rate Law: $ln[A]_t = -kt + ln[A]_0$ or $[A]_t = [A]_0e^{-kt}$
  • Half-Life: $t_{1/2} = \frac{0.693}{k}$
• Second-Order Reactions
  • Rate Law: $Rate = k[A]^2$
  • Integrated Rate Law: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$
  • Half-Life: $t_{1/2} = \frac{1}{k[A]_0}$
• $[A]_t$ is the concentration of $A$ at time $t$.
• $[A]_0$ is the initial concentration of $A$.

Collision Theory

• For a reaction to occur, molecules must collide with the correct orientation and sufficient energy.

Transition State Theory

• Transition State: Highest-energy state in a reaction pathway, also called an activated complex.
• Reactions proceed through a transition state that requires overcoming an activation energy barrier.