Chapter2-Numpy Flashcards
Dot product
v transpose dot w
np.dot(v.T, w)
v.T.shape
get the shape (number of lines, number of columns) of v transpose
how to create a 2x2 Matrix A as 2D Numpy Array having the elements 1234?
A = np.array([[1,2],[3,4]])
create a vector vdont with shape(3,). Is it a mathematical vector?
vdont = np.array([1,1,1])
if x is a numpy array([[9]]) what is the output of print(x.item())
when that array contains exactly one item results on this element: 9
how to get the norm of a vector v?
np.linalg.norm(v)
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?
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)
What is the definition of the identity matrix? How to create one identity matrix I3 of grad 3?
definition: Matrix with all elements in the main diagonal set to 1. I3= np.identity(3)
what is the result np.dot(I3, w3) for all vectors w3 of shape (3,*)?
the vector w3 again
How is the inverse of a Matrix defined and when does the inverse of a matrix exist?
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:
- Square Matrix: The matrix must be a square matrix, which means it has the same number of rows and columns
- 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.
- 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.
Given a system of equations in the form Ax = b, how to solve this system for x mathematicaly and in python?
math: x = A⁻¹ * b
python: x = np.dot(np.linalg.inv(A), b)
what is the diference between np.dot(A, B) and A * B?
A * B ist the element-wise multiplication
How to obtain the type of data stored in one array A?
A.dtype
how to create one column vector np array containing 10 equally spaced real number from 1 to 5?
np.linspace(1,5,10).reshape(10,1)
what do the function calls np.hstack([v,w]) and np.vstack([v,w]) do?
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
what to get
array([[1, 2, 1, 2, 1, 2],
[3, 4, 3, 4, 3, 4]])
from
A = array([[1, 2],
[3, 4]])
np.tile(A, 3)
how to create one vector containing the natural numbers from 0 to 29?
np.arange(30)
how to get the first column of a matrix?
A[:, 0]
which elements o the lines and columns of A define the new matrix extracted by view1 = A[0:3, 0:3]
line: elements 0, 1 and 2
column: elements 0, 1 and 2
considering isBo to be array([ True, False, False, True, False, True, True]) what does data[isBo, :] will retrieve?
True for every one that I want and false for every row I don’t want to extract
if data was defined as np.arrange(0, 28), how to get all the even numbers 0, 2, … , 26 ?
data[data%2==0]
how to directly solve the system of linear equation A.x-> = b-> for x?
x will be np.linalg.solve(A, b)
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?
the result is np.array([[4,6],[3,5]] because o broadcasting
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?
- rgn = np.random.default_rgn()
- X = rgn.standard_normal(size=(4,3))
- col_means = X.mean(0)
- X * np.sqrt(5) + 4
what is the difference between X.mean(axis=0) and X.mean(axis=1)?
- 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
Replace all odd numbers in arr x with -1 without changing x and store in y.
y = np.where(x%2 == 1, -1, x)
how to get from (nxm) matrix X to non math array with shape (n*m, )?
X.flatten()
how to apply one function in whole np array x?
- 1 - define the function
- 2 - vectorize de function: vectorizedfunction = np.vectorize(function)
- 3 - call the vectorized function on the np array x : vectorizedfunction(x)
how to sum all the elements of np array x?
np.add.reduce(x)
how to multiply over all elements of np array x?
np.multiply.reduce(x)
what are usually axis 0 and 1 in np?
axis 0 = lines, axis 1 = columns
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,)?
- 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)
how to get the number of dimension of a matrix m?
m.ndim
what is the requirement to apply reshape?
the new shape must contain the same number of entries ( e.g. rows* columns)
how to swipe the second and third columns of a matrix u ? Explain the synthax.
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
if a vector v has shape (3,) what is the shape of v.transpose() ?
(3,)
what does the method .all() do?
returns true if all elements of iterable are true
what is the output of follow code and why?
v3 = np.array([‘a’, 1, 5.0])
print(v3[1] + v3[2])
15.0 => because of the ‘a’ v3 stores strings and the operator + turns to concatenation
Is np.array([2,3,4,5,6,7]).reshape(3,2).T = np.array([2,3,4,5,6,7]).reshape(2,3) ?
NO!
O que significa ter uma Metriz M que é tranformacao linear que leva um ponto P ate um ponto R? Como
M . P = R
como mapear resolver A . P = b, conhecendo P e b:
- se b for invertable
- se b nao for invertable
- A = np.dot(b, np.linalg.inv(P))
- A = np.dot(b, np.linalg.pinv(P))
How to determine the rank of a matrix ?
np.linalg.matrix_rank
como calcular sin e cos?
sin_values = np.sin(angle)
cos_values = np.cos(angle)
what does apply rotation r to a cube mean in terms of dot product?
cube . r
how to use np.save and np.load ?
-np save: np.save(‘array.npy’, M)
-np load: array_load = np.load(‘array.npy’)
hot to find the index of max value in an array?
np.argmax(array)
which numbers are contained in the array np.arange(-1000, 1000)
intergers [-1000, -999, …, 998, 999]
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?
Nao. s = s.reshape(2, 4) cria uma copia e nao uma view
o que é uma fatias simples? o que é uma fatia avancada? Qual é a diferenca em termos de copia/view?
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
df.loc[row_start:row_end, col_name] row_end é incluido ou nao?
sim. Para “loc” a selecao é inclusiva para o inicio e o fim.
