[SOLVED] DataScience -Data Science - Assignment 3 -Matrix fun

Description

Solve the following using Python with numpy.

In numpy, there are some handy ways of working with matrixes (they are all explained later in this document):

import numpy as np from numpy.linalg import inv # Creating matrices A   = np.array([[ 1, 2 ],[ 3, 4]]) B   = np.array([[ 9, 8 ],[ 7, 6]]) # Transposing: A.T # A transposed (danish: A transponeret) B.T # B transposed # Matrix multiplication: A @ B # Inverse: inv(A)

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Figure 1: Basic matrix functionality with numpy.

In the following, when talking about multiplication, we implicitly mean matrix multiplication (same as dot product).

Task 1

Given the two following matrices

• Find A^T
• Find B^T
• Find AB (matrix multiplication). Compare with simple multiplication (using * instead of @ in Python). Can you see what is the difference?
• Find AB^T
• Compare AB^T and B^TA^T
• Find (A^T)^T
• Find AA^T

Task 2

Given

• Find AB
• Find BA

Confirm that they are different! Clearly, when doing matrix multiplication, order matters! AB 6= BA, so matrix multiplication is not commutative.

Task 3

The inverse of a matrix  is found by

(1)

As seen in listing 1, the inverse of a matrix can be found easily with numpy (after having imported numpy.linalg.inv) by: inv(A). Using the same matrices from Task 2:

• Find A⁻¹
• Find B⁻¹
• Find AA⁻¹. Look closely at the result. (d) Find A⁻¹A. Look closely at the result.
• Find BB⁻¹. Look closely at the result.
• Find B⁻¹B. Do you start to see a pattern?

It appears that a matrix multiplied with its inverse always gives .

Incidentally, a matrix with only ones in the diagonal is called an identity matrix, often denoted I.

Task 4

Given

(a) Find A⁻¹

Oops. We see that not all matrices have an inverse! Looking at equation 1 (the equation for finding the inverse), can you figure out why? (hint: look at the denominator!)

Task 5

Plotting (lines, graphs, coordinates, etc) can be done using matplotlib. Try the following:

import numpy as np from matplotlib import pyplot as plt from numpy.linalg import inv xs = np.array([0,0,3,3,0,1.5,3]) # List of x coordinates ys = np.array([1,0,0,1,1,2 ,1]) # List of y coordinates fig = plt.figure() fig, ax = plt.subplots() xs_ys = np.array([xs,ys]) ax.axis(’equal’) # Plot the points ax.plot( *xs_ys, ’-’, color=’b’) # Create a rotation matrix rot = np.array([[1, 0],[0, 1]]) # <– CHANGE THIS # turn the two lists (xs, ys) into a list of (x,y) tuples points = np.array([[x,y] for x,y in zip(xs,ys)]) # Make the transformation: points_rot = (points @ rot) # Turn it into a row of xs and a row of ys: xs_ys_rot = np.array([points_rot[:,0], points_rot[:,1]]) # Finally, plot it ax.plot( *xs_ys_rot, ’-’, color=’r’) fig

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(a) Set the rotation matrix (line 18) to rotate the shape 45 degrees ( radians).