Exercise 8.2: Working with two-dimensional arrays

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import numpy as np

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Numpy enables you do to matrix calculations on two-dimensional arrays. In exercise, you will practice doing matrix calculations on arrays. We’ll start by making a matrix and a vector to practice with. You can copy and paste the code below.

A = np.array(
    [
        [6.7, 1.3, 0.6, 0.7],
        [0.1, 5.5, 0.4, 2.4],
        [1.1, 0.8, 4.5, 1.7],
        [0.0, 1.5, 3.4, 7.5],
    ]
)

b = np.array([1.1, 2.3, 3.3, 3.9])

a) First, let’s practice slicing.

1. Print row 1 (remember, indexing starts at zero) of A.

2. Print columns 1 and 3 of A.

3. Print the values of every entry in A that is greater than 2.

4. Print the diagonal of A. using the np.diag() function.

b) The np.linalg module has some powerful linear algebra tools.

1. First, we’ll solve the linear system \(\mathsf{A}\cdot \mathbf{x} = \mathbf{b}\). Try it out: use np.linalg.solve(). Store your answer in the Numpy array x.

2. Now do np.dot(A, x) to verify that \(\mathsf{A}\cdot \mathbf{x} = \mathbf{b}\).

3. Use np.transpose() to compute the transpose of A.

4. Use np.linalg.inv() to compute the inverse of A.

c) Sometimes you want to convert a two-dimensional array to a one-dimensional array. This can be done with np.ravel().

1. See what happens when you do B = np.ravel(A).

2. Look of the documentation for np.reshape(). Then, reshape B to make it look like A again.

Solution

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a)

# 1.
print('first row of A:\n', A[1,:])

# 2.
print('\ncolumns 1 and 3 of A:\n', A[:,[1,3]])

# 3.
print('\nvalues of every entry in A that is greater than 2:\n', A[A>2])

# 4.
print('\ndiagonal of A:\n', np.diag(A))

first row of A:
 [0.1 5.5 0.4 2.4]

columns 1 and 3 of A:
 [[1.3 0.7]
 [5.5 2.4]
 [0.8 1.7]
 [1.5 7.5]]

values of every entry in A that is greater than 2:
 [6.7 5.5 2.4 4.5 3.4 7.5]

diagonal of A:
 [6.7 5.5 4.5 7.5]

b)

# 1.
x = np.linalg.solve(A, b)

# 2.
print('A . x:\n', np.dot(A, x))
print('\nb:\n', b)

# 3.
print('\ntranspose of A:\n', np.transpose(A))

# 4.
print('\ninverse of A:\n', np.linalg.inv(A))

A . x:
 [1.1 2.3 3.3 3.9]

b:
 [1.1 2.3 3.3 3.9]

transpose of A:
 [[6.7 0.1 1.1 0. ]
 [1.3 5.5 0.8 1.5]
 [0.6 0.4 4.5 3.4]
 [0.7 2.4 1.7 7.5]]

inverse of A:
 [[ 0.15267508 -0.03365026 -0.01778     0.00054854]
 [-0.00906001  0.19788853  0.03719385 -0.07090934]
 [-0.04391535 -0.0144834   0.26880108 -0.05219479]
 [ 0.02172029 -0.0330119  -0.12929526  0.17117684]]

c)

# 1.
B = np.ravel(A)
print('result of np.ravel(A):\n', B)

# 2.
A_from_reshape = np.reshape(B, A.shape)
print('\nA from reshaped B:\n', A_from_reshape)

result of np.ravel(A):
 [6.7 1.3 0.6 0.7 0.1 5.5 0.4 2.4 1.1 0.8 4.5 1.7 0.  1.5 3.4 7.5]

A from reshaped B:
 [[6.7 1.3 0.6 0.7]
 [0.1 5.5 0.4 2.4]
 [1.1 0.8 4.5 1.7]
 [0.  1.5 3.4 7.5]]

Computing environment

%load_ext watermark
%watermark -v -p numpy,jupyterlab

CPython 3.7.7
IPython 7.16.1

numpy 1.18.5
jupyterlab 2.1.5