Python Problem 70

Question

Use Python:

Define the following vectors and matrices:

vec1 = np.array([ -1., 4., -9.])

mat1 = np.array([[ 1., 3., 5.], [7., -9., 2.], [4., 6., 8. ]]

1. You can multiply vectors by constants. Compute

vec2 = (np.pi/4) * vec1

2. The cosine function can be applied to a vector to yield a vector of cosines.

Compute

vec2 = np.cos( vec2 )

3. You can add vectors and multiply by scalars. Compute

vec3 = vec1 + 2 * vec2

4. The Euclidean norm of a matrix or a vector is available using la.norm.

Compute

la.norm(vec3)

5. You can do row-column matrix multiplication. Compute the product of

mat1 and vec3 and set vec4 equal to the result.

6. Compute the transpose of mat1.

7. Compute the determinant of mat1.

8. Compute the trace of mat1.

9. Find the smallest element in vec1.

Answer 1

import numpy as np

import numpy.linalg as la

# Define the vectors and matrices

vec1 = np.array([-1., 4., -9.])

mat1 = np.array([[1., 3., 5.], [7., -9., 2.], [4., 6., 8.]])

# 1. Multiply vec1 by a constant

vec2 = (np.pi/4) * vec1

print("vec2 = ", vec2)

# 2. Apply cosine function to vec2

vec2 = np.cos(vec2)

print("vec2 = ", vec2)

# 3. Add vectors and multiply by scalars

vec3 = vec1 + 2 * vec2

print("vec3 = ", vec3)

# 4. Compute Euclidean norm of vec3

norm_vec3 = la.norm(vec3)

print("norm_vec3 = ", norm_vec3)

# 5. Matrix-vector multiplication

vec4 = np.dot(mat1, vec3)

print("vec4 = ", vec4)

# 6. Transpose of mat1

mat1_transpose = mat1.T

print("mat1_transpose = ", mat1_transpose)

# 7. Determinant of mat1

mat1_det = la.det(mat1)

print("mat1_det = ", mat1_det)

# 8. Trace of mat1

mat1_trace = np.trace(mat1)

print("mat1_trace = ", mat1_trace)

# 9. Smallest element in vec1

smallest_elem_vec1 = np.min(vec1)

print("smallest_elem_vec1 = ", smallest_elem_vec1)