Chapter2-Numpy

Question 1

Dot product
v transpose dot w

Answer 1

np.dot(v.T, w)

Question 2

v.T.shape

Answer 2

get the shape (number of lines, number of columns) of v transpose

Question 3

how to create a 2x2 Matrix A as 2D Numpy Array having the elements 1234?

Answer 3

A = np.array([[1,2],[3,4]])

Question 4

create a vector vdont with shape(3,). Is it a mathematical vector?

Answer 4

vdont = np.array([1,1,1])

Question 5

if x is a numpy array([[9]]) what is the output of print(x.item())

Answer 5

when that array contains exactly one item results on this element: 9

Question 6

how to get the norm of a vector v?

Answer 6

np.linalg.norm(v)

Question 7

matrix A with shape (3,2) and vector u with shape (2,1). Is it possible in Python to calculate A dot u? If yes, what is the shape of the result?

Answer 7

Yes because the number of columns of the matrix is equal the number of elements of the vector. The resulting vector will have shape (3,1)

Question 8

What is the definition of the identity matrix? How to create one identity matrix I3 of grad 3?

Answer 8

definition: Matrix with all elements in the main diagonal set to 1. I3= np.identity(3)

Question 9

what is the result np.dot(I3, w3) for all vectors w3 of shape (3,*)?

Answer 9

the vector w3 again

Question 10

How is the inverse of a Matrix defined and when does the inverse of a matrix exist?

Answer 10

Mathematically, the inverse of a matrix A is denoted as A⁻¹, and it satisfies the equation:

A * A⁻¹ = A⁻¹ * A = I

follow criteria must be met:

1. Square Matrix: The matrix must be a square matrix, which means it has the same number of rows and columns
2. Full Rank: The matrix must be of full rank. A matrix is said to be of full rank if its rows and columns are linearly independent.
3. Non-Singular (or Non-Degenerate): A matrix is invertible if and only if it is non-singular (or non-degenerate). A non-singular matrix is one that does not have a determinant of zero.

Question 11

Given a system of equations in the form Ax = b, how to solve this system for x mathematicaly and in python?

Answer 11

math: x = A⁻¹ * b
python: x = np.dot(np.linalg.inv(A), b)

Question 12

what is the diference between np.dot(A, B) and A * B?

Answer 12

A * B ist the element-wise multiplication

Question 13

How to obtain the type of data stored in one array A?

Answer 13

A.dtype

Question 14

how to create one column vector np array containing 10 equally spaced real number from 1 to 5?

Answer 14

np.linspace(1,5,10).reshape(10,1)

Question 15

what do the function calls np.hstack([v,w]) and np.vstack([v,w]) do?

Answer 15

np.hstack([v,w]): attach the vector w as a column to the vector v creting a matrix
np.vstack([v,w]): attach the vector w as a ‘line’ to the vector v creting a matrix

Question 16

what to get
array([[1, 2, 1, 2, 1, 2],
[3, 4, 3, 4, 3, 4]])
from
A = array([[1, 2],
[3, 4]])

Answer 16

np.tile(A, 3)

Question 17

how to create one vector containing the natural numbers from 0 to 29?

Answer 17

np.arange(30)

Question 18

how to get the first column of a matrix?

Answer 18

A[:, 0]

Question 19

which elements o the lines and columns of A define the new matrix extracted by view1 = A[0:3, 0:3]

Answer 19

line: elements 0, 1 and 2
column: elements 0, 1 and 2

Question 20

considering isBo to be array([ True, False, False, True, False, True, True]) what does data[isBo, :] will retrieve?

Answer 20

True for every one that I want and false for every row I don’t want to extract

Question 21

if data was defined as np.arrange(0, 28), how to get all the even numbers 0, 2, … , 26 ?

Answer 21

data[data%2==0]

Question 22

how to directly solve the system of linear equation A.x-> = b-> for x?

Answer 22

x will be np.linalg.solve(A, b)

Question 23

What is the result of the operation np.array([[3],[2]]) with shape (2,1) and np.array([[3, 2]]) with shape (1,2) ? why?

Answer 23

the result is np.array([[4,6],[3,5]] because o broadcasting

Question 24

create one object of random number generator on the variable rgn. Use rgn to generate one matrix of standard normal distributed numbers of shape (4,2) and store in col_means the vector of column means.
How to transform the matrix onto mean = 4, variance=5?

Answer 24

• rgn = np.random.default_rgn()
• X = rgn.standard_normal(size=(4,3))
• col_means = X.mean(0)
• X * np.sqrt(5) + 4

Question 25

what is the difference between X.mean(axis=0) and X.mean(axis=1)?

Answer 25

• X.mean(axis=0) => jump along the axis of rows, calculating the mean for each column
• X.mean(axis=1) => jump along the axis of columns, calculating the mean for each row

Question 26

Replace all odd numbers in arr x with -1 without changing x and store in y.

Answer 26

y = np.where(x%2 == 1, -1, x)

Question 27

how to get from (nxm) matrix X to non math array with shape (n*m, )?

Answer 27

X.flatten()

Question 28

how to apply one function in whole np array x?

Answer 28

• 1 - define the function
• 2 - vectorize de function: vectorizedfunction = np.vectorize(function)
• 3 - call the vectorized function on the np array x : vectorizedfunction(x)

Question 29

how to sum all the elements of np array x?

Answer 29

np.add.reduce(x)

Question 30

how to multiply over all elements of np array x?

Answer 30

np.multiply.reduce(x)

Question 31

what are usually axis 0 and 1 in np?

Answer 31

axis 0 = lines, axis 1 = columns

Question 32

explain how numpy compaires shapes of elements to possible broadcasting them? Does both elements need to have same number of dimensions? What happens with missing dimensions like broadcasting (8,2) and (8,)?

Answer 32

• element-wise starting from the rightmost working its way left. Two dim are compatible when they are equal or one of them are 1.
• The resulting array will have the same number of dimensions as the input array with the greatest number of dimensions, where the size of each dimension is the largest size of the corresponding dimension among the input arrays.
• broadcasting (8,2) and (8,) will not work because (8,) will be threated like (1,8)

Question 33

how to get the number of dimension of a matrix m?

Answer 33

m.ndim

Question 34

what is the requirement to apply reshape?

Answer 34

the new shape must contain the same number of entries ( e.g. rows* columns)

Question 35

how to swipe the second and third columns of a matrix u ? Explain the synthax.

Answer 35

u[:,[1,2]] = u[:,[2,1]]
creates a view where all rows and second, third columns is assigned to all rows and third, second columns

Question 36

if a vector v has shape (3,) what is the shape of v.transpose() ?

Answer 36

(3,)

Question 37

what does the method .all() do?

Answer 37

returns true if all elements of iterable are true

Question 38

what is the output of follow code and why?
v3 = np.array([‘a’, 1, 5.0])
print(v3[1] + v3[2])

Answer 38

15.0 => because of the ‘a’ v3 stores strings and the operator + turns to concatenation

Question 39

Is np.array([2,3,4,5,6,7]).reshape(3,2).T = np.array([2,3,4,5,6,7]).reshape(2,3) ?

Answer 39

NO!

Question 40

O que significa ter uma Metriz M que é tranformacao linear que leva um ponto P ate um ponto R? Como

Answer 40

M . P = R

Question 41

como mapear resolver A . P = b, conhecendo P e b:
- se b for invertable
- se b nao for invertable

Answer 41

• A = np.dot(b, np.linalg.inv(P))
• A = np.dot(b, np.linalg.pinv(P))

Question 42

How to determine the rank of a matrix ?

Answer 42

np.linalg.matrix_rank

Question 43

como calcular sin e cos?

Answer 43

sin_values = np.sin(angle)
cos_values = np.cos(angle)

Question 44

what does apply rotation r to a cube mean in terms of dot product?

Answer 44

cube . r

Question 45

how to use np.save and np.load ?

Answer 45

-np save: np.save(‘array.npy’, M)
-np load: array_load = np.load(‘array.npy’)

Question 46

hot to find the index of max value in an array?

Answer 46

np.argmax(array)

Question 47

which numbers are contained in the array np.arange(-1000, 1000)

Answer 47

intergers [-1000, -999, …, 998, 999]

Question 48

a é uma matriz 4x5 e s é uma view das colunas 1 e 2 de a. Apos s = s.reshape(2, 4), a ta,be, sera modificado?

Answer 48

Nao. s = s.reshape(2, 4) cria uma copia e nao uma view

Question 49

o que é uma fatias simples? o que é uma fatia avancada? Qual é a diferenca em termos de copia/view?

Answer 49

fatia simples: array[start:stop:step], cria view
fatia avancada: Utiliza índices avançados, como listas de índices, fatias não contínuas ou máscaras booleanas., cria copia

Question 50

df.loc[row_start:row_end, col_name] row_end é incluido ou nao?

Answer 50

sim. Para “loc” a selecao é inclusiva para o inicio e o fim.