Linear Algebra Technical

Question 1

Initialize a vector

Answer 1

import numpy as np
np.array([1,2,3]) or np.array([1], [2], [3])

Question 2

Calculate the magnitude of a vector

Answer 2

np.linalg.norm(v)

Question 3

Calculate the direction of a vector

Question 4

Add two vectors

Answer 4

v1 + v2

Question 5

Scale a vector

Answer 5

3 * v1

Question 6

Subtract two vectors

Answer 6

v1 - v2

Question 7

Initialize a zero vector

Answer 7

np.zeros(3)

Question 8

Get the negative of a vector

Answer 8

-1 * v1

Question 9

Perform dot product of two vectors

Answer 9

1. numpy.matmul(v1, v2)
2. v1.dot(v2)

Question 10

Create a random vector

Answer 10

np.random.rand(3,1)

Question 11

Calculate the cross product of a vector

Answer 11

a = np.array([1,2,3])
b = np.array([4,5,6])
a.shape # (3,)
b.shape # (3.)
np.cross(a,b)

Question 12

What is the angle between two vectors

Answer 12

np.arccose(np.dot(a1, a2)/ np.linalg(a1) * np.linalg.norm(a2))

Question 13

Get the Euclidian distance of a vector

Answer 13

point1 = np.array((1, 2, 3))
point2 = np.array((1, 1, 1))
dist = np.linalg.norm(point1 - point2)

Question 14

Get the projection of a vector

Answer 14

Projection of the vector = a.b/||b||**2 * b
scalar_projection = np.dot(a,b)/ np.dot(b,b)
vector_projection = scalar_projection * b

Question 15

Create linearly independent matrix

Answer 15

np.random.randint(10, size=(2,5))

Question 16

Perform dot product of orthogonal vectors

Answer 16

a1 = np.array([2,3])
a2 = np.array([3, -2])
dot(a1,a2)

Question 17

Initialize a matrix

Answer 17

import numpy as np

rows = 3
cols = 2
size = rows*cols

mat = np.array([0]*size).reshape(rows,cols)

Question 18

Calculate matrix determinant

Answer 18

mat = np.random.randint(25, size=(5,5))
determinant = np.linalg.det(mat)

Question 19

Prove invertible matrix vs det

Question 20

Linearly dependent vs det

Question 21

Prove properties of det A = 0

Question 22

Calculate rank of a matrix - different approaches

Question 23

Calculate rank of a matrix -

Question 24

Transpose a matrix

Question 25

Eigen value calcualtion

Question 26

Eigen vector calculation

Question 27

Create unit vector