Linear Algebra Python

Question 1

Initialize a vector

Answer 1

1. v = [3,2]
2. v1 = np.array([3,2])

Question 2

Addition of two vectors

Answer 2

v1 + v2

Question 3

Scalar multiplication

Answer 3

2 * v

Question 4

Transpose of a vector

Answer 4

1. np.transpose(v)
2. v.T
3. v.T.T

Question 5

Dot product approaches

Answer 5

1. np.dot(v1, v2)
2. np.matmul(v1, v2)
3. sum(np.multiply(v1, v2))

Question 6

Create one random number

Answer 6

np.random.rand/ np.random.randn

Question 7

Create random vector of size 10

Answer 7

np.random.randn(10)

Question 8

create a 3 * 1 matrix of random numbers

Answer 8

np.random.randn(3,1)

Question 9

Get the dimensions of a matrix

Answer 9

mat.ndim

Question 10

Get the number of elements in a matrix

Answer 10

mat.size

Question 11

Get the size of the matrix - 4X6

Answer 11

mat.shape

Question 12

Just get the number of rows of a matrix

Answer 12

mat.shape[0]

Question 13

Just get the number of columns of a matrix

Answer 13

mat.shape[1]

Question 14

Get the first column of a matrix

Answer 14

mat[: , 0]

Question 15

Get the first and third columns of a matrix

Answer 15

mat[: , [0,2]]

Question 16

Get the one to third columns of a matrix

Answer 16

mat[: , 0:3]

Question 17

Repeat 1st column twice in a matrix

Answer 17

mat[:, [0,0]]

Question 18

Get the first row of a matrix

Answer 18

mat[0, :]

Question 19

Get the first and third rows of a matrix

Answer 19

mat[[0,2], :]

Question 20

Get 1 to third rows in a matrix

Answer 20

mat[0:3 , :]

Question 21

Repeat 1st row twice in a matrix

Answer 21

mat[[0,0], :]

Question 22

Repeat 1st row n times in a matrix

Answer 22

mat[[0] * n , :]

Question 23

Create a zero vector of size 4

Answer 23

np.zeros(4) # 4 row vector

Question 24

convert 3X1 matrix to 3x1 vector

Answer 24

1. mat = np.random.randn(3,1)
2. arr = np.asarray(mat) # 2D array
3. vector = np.squeeze(arr) # 1d array

Question 25

magnitude of a vector

Answer 25

1. np.linalg.norm(v)
2. np.sqrt(sum(np.multiply(v,v)))
3. np.sqrt(np.dot(v,v))

Question 26

Angle between two vectors

Answer 26

theta = np.arccos((np.dot(a,b)) / (np.linalg.norm(a) * np.linalg.norm(b)))

Question 27

How to know if two vectors are orthogonal

Answer 27

If the dot product of the two vectors is 0

Question 28

How to create just positive random number

Answer 28

np.random.rand()

Question 29

create duplicate dependent vector from v1

Answer 29

np.random.rand() * v

Question 30

absolute value of a number

Answer 30

np.abs(-1)

Question 31

Perform hadamard multiplication

Answer 31

1. np.multiply(v1, v2)
2. v1 * v2

Question 32

Outer product

Answer 32

np.outer(v1, v2)

Question 33

cross product of vectors

Answer 33

np.cross(v1, v2)

Question 34

create complex number

Answer 34

np.complex(3,4) = 3 + 4j

Question 35

Create complex vector

Answer 35

np.array([3, 4i, np.complex(5,2), 2-5i])

Question 36

np.transpose(3 + 4i) =

Answer 36

3 + 4i

Question 37

Hermitian transpose of v = 3+4i - write syntax

Answer 37

np.transpose(v.conjugate()) * v

Question 38

Unit vectors - mu =

Answer 38

1/np.linalg.norm(v)

Question 39

Normalize a vector

Answer 39

mu * v= norm_vector
np.linalg.norm(norm_vector) = 1

Question 40

Scalar multiplication

Answer 40

2 * v

Question 41

Creating an identity matrix

Answer 41

np.eye(3), np.eye(3,5); But identity matrices are only square matrices in practice

Question 42

Creating zeros matrix

Answer 42

np.zeros((3,3)), np.zeros((3,5))

Question 43

creating diagonal matrix

Answer 43

np.diag([1,2,3,4,5]); Can create rectangular matrices as well

Question 44

Creating upper triangular matrix

Answer 44

s = np.random.randn(5,5)
ut = np.triu(s); Can create rectangular matrices as well

Question 45

Creating lower triangular matrix

Answer 45

s = np.random.randn(5,5)
tl = np.tril(s); Can create rectangular matrices as well

Question 46

Concatenate two matrices

Answer 46

np.concatenate((A,B), axis = 1)

Question 47

Addition of matrices

Answer 47

A + B; Both the matrices must be of same size

Question 48

Shifting a matrix

Answer 48

d = np.random.randn(3,3)
si = 0.3 * eye(3)
shiftedD = d + si

Question 49

Create 2x2 matrix

Answer 49

np.array([[2,3], [3,4]])

Question 50

Convert np array to matrix

Answer 50

c = np.array([1, 2+9j, 3, 4])
m = np.matrix(c)

Question 51

Transposes on matrix

Answer 51

1. np.matrix(c).H
2. np.matrix(c).T

Question 52

Two applications of diag keyword

Answer 52

1. d = np.diag(matrix) - creates a row vector with the diagonal elements as row vector
2. x = np.diag(d) - creates a diagonal matrix with the elements of the row vector as diagonal elements.

Question 53

Calculate trace

Answer 53

1. np.trace(matrix)
2. np.sum(np.diag(matrix)))
3. sum(np.diag(matrix)))

Question 54

Properties that prove trace is a linear operator

Answer 54

1. np.trace(a+b) = np.trace(a) + np.trace(b)
2. l * np.trace(a) = np.trace(l*a)
   closed under both addition and subtraction

Question 55

Create a matrix with values from 1 to 12 and shape 3X4

Answer 55

a = np.reshape(np.arange(1,13), (3,4), ‘F’)

Question 56

Third parameter in np.reshape

Answer 56

F - column
C - row - default

Question 57

Broadcasting row vectors

Answer 57

m = np.reshape(np.nrange(1,13), (3,4), ‘F’)
a = np.array([10, 20, 30, 40])
m+a

Question 58

Broadcasting column vectors

Answer 58

a + r
a+ c

Question 59

Creating a symmetric matrix from a non-symmetric matrix a

Answer 59

a * a.T/m ** 2

Question 60

Create a random integer matrix

Answer 60

np.random.randint(1,12, (m,m))

Question 61

Hadamard matrix multiplication

Answer 61

1. np.multiply( a, b)
2. a*b
   a@b - regular matrix multiplication

Question 62

Hadamard multiplication rule

Answer 62

Both the matrices must be of same size