3.2. Chapter 2: Intro to Linear Algebra#

Before Class:

During and After Class:

  • Take notes (on paper or a tablet computer)

  • Complete this notebook, submit you answer via Gradescope

Other Resources:

import matplotlib.pyplot as plt
import numpy as np
import scipy.linalg as la

3.2.1. Identify Linear Functions#

P2.1 in Savov (2020)

Classify the following functions as linear or nonlinear?

  1. \(q(x) = x^2\)

  2. \(f(x) = g(h(x))\), where \(g(x) = \sqrt{3}~x\) and \(h(x) = -4 x\)

  3. \(i(x) = \frac{1}{m~x}\)

  4. \(j(x) = \frac{x-a}{x-b}\)

For each function, first determine if it is linear using pencil and paper to test your intuition (do not turn in – honor system). Next, plot the function using Python. Finally, write a few words stating if the function is nonlinear or not and justifying your answer.

# Let's evaluate x from -5 to 5 for ploting
x = np.linspace(-5,5,21)

# function 1
f1 = lambda x : x**2
plt.plot(x, f1(x))
plt.xlabel(r'$x$',fontsize=16)
plt.ylabel(r'$q(x) = x^2$',fontsize=16)
plt.show()

# Add your solution here

Your Answers:

  1. Fill in

  2. Fill in

  3. Fill in

  4. Fill in

3.2.2. Vector Operations#

Given the vectors, \({\bf u} = (1,1,1)\), \({\bf v} = (2,3,1)\), \({\bf w} = (-1,-1,2)\), we will compute several products.

First, let’s plot the vectors in 3D with Python.

# define vectors
u = np.array([1,1,1])
v = np.array([2,3,1])
w = np.array([-1,-1,2])

# Create 3D figure and axes
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')

# Set labels and limits
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('z')
ax.set_xlim([-1,2])
ax.set_ylim([-1,3])
ax.set_zlim([0,2])

# Plot vectors
def plot_vector(vector, color, label):
    ''' Plot a 3D vector
        Arguments:
            vector: np array
            color: 'r', 'b', 'g', etc.
            label: string for legend
        
    '''
    assert len(vector) == 3, "Must be a 3D vector"

    ax.quiver(0, 0, 0, vector[0], vector[1], vector[2], color=color, arrow_length_ratio=0.1, label=label)

plot_vector(u, 'r', r'${\bf u}$')
plot_vector(v, 'b', r'${\bf v}$')
plot_vector(w, 'g', r'${\bf w}$')

plt.legend(loc='best',ncol=3)
plt.show()

3.2.2.1. Dot Products#

Compute these products by hand (do not turn in – honor system) and with Python.

  1. \({\bf u} \cdot {\bf v}\)

  2. \({\bf u} \cdot {\bf w}\)

  3. \({\bf v} \cdot {\bf w}\)

# Add your solution here

Uses the 3D visualization of these vectors, explain your answers for the three dot products in a few sentences. Hint: what is the geometric interpretation of a dot product?

Your Explanation:

3.2.2.2. Cross Products#

Compute these products by hand (do not turn in – honor system) and with Python.

  1. \({\bf u} \times {\bf v}\)

  2. \({\bf u} \times {\bf w}\)

  3. \({\bf v} \times {\bf w}\)

# Add your solution here

# Add your solution here

# Add your solution here

Uses the 3D visualization of these vectors, explain your answers for the three cross products in a few sentences. Hint: what is the geometric interpretation of a cross product?

Your Answer:

3.2.3. Linear Transformations#

Reference: P2.13 in Savov (2020).

Consider the following three linear transformations that take a two dimensional vector as an input and output a two dimensional vector.

\[\begin{split} T_A \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} 3x \\ y \end{bmatrix}, \quad T_B \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} x + y \\ y \end{bmatrix}, \quad T_C \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} 3x + y \\ y \end{bmatrix} \end{split}\]

3.2.3.1. Corresponding Matrixes#

Find the 2 x 2 matrices \(\bf A\), \(\bf B\), and \(\bf C\) such that

\[ T_A({\bf v}) = {\bf A v}, \quad T_B({\bf v}) = {\bf B v}, \quad T_C({\bf v}) = {\bf C v}, \quad \forall {\bf~v} \]

Hint: Do this on paper first!

# Define your answer as matrices/arrays in numpy.
# Store in variable names A, B, and C
# We'll use this for the next step

# Add your solution here

print("A = \n",A)
print("\nB = \n",B)
print("\nC = \n",C)

3.2.3.2. Matrix Multiplication#

Calculate the matrix products \(\bf A B\) and \(\bf B A\) first on paper and then with Python. (Yes, you should practice doing matrix multiplication by hand at least once.)

# Store your answers in the variables AB and BA
AB = A @ B

# Add your solution here

print("\nAB =\n",AB)
print("\nBA =\n",BA)

Based on your calculations, is matrix multiplication commutative? Explain in a sentence.

Your Answer:

How is \({\bf C}\) related to \({\bf A}\) and \({\bf B}\)? Explain why this makes sense using the word “composite”.

Your Answer:

3.2.4. Atomic Species Balance using Linear Algebra#

Reference: Example 4.7-1 in Elementary Principles of Chemical Process Engineering, Third Edition, Felder and Rousseau (2005)

Methane is burned with air in a continuous steady-state combustion reactor to yield a mixture of carbon monoxide, carbon dioxide, and water. The reactions taking place are:

CH\(_4\) + \(\frac{3}{2}\) O\(_2\) \(\rightarrow\) CO + 2 H\(_2\)O

CH\(_4\) + 2 O\(_2\) \(\rightarrow\) CO\(_2\) + 2 H\(_2\)O

The feed to the reactor contains 7.80 mole% CH\(_4\), 19.4% O\(_2\), and 72.8% N\(_2\). The percentage conversion of methane is 90.0%, and the gas leaving the reactor contains 8 mol CO\(_2\)/mol CO. What is the molar composition of the product stream?

Do the following:

  1. Draw a picture (on paper)

  2. Propose a mathematical model using atomic species balances (on paper)

  3. Perform degree of freedom analysis (on paper)

  4. Convert your model to a linear system (on paper)

  5. Solve with Python (in this notebook)

  6. Print your answer to the screen with a reasonable number of significant digits (in this notebook)

# Add your solution here

3.2.5. Submission#

Please submit two files:

  1. Submit your pencil and paper calculations and pseudocode as requested to Gradescope (one assignment)

  2. Download this notebook as a .ipynb file and submit to to Gradescope (second assignment)

Reminders:

  • Be sure to upload your pencil and paper calculations and pseudocode to Gradescope as a single PDF for grading.

  • Review the homework formatting guidelines in the syllabus.

  • Review the commenting and pseudocode guidlines.