# Day 8 exercise solutions¶

## Simple arrays¶

5 minutes.

Create this array:

2  7 12  0
3  9  3  4
4  0  1  3

>>> import numpy as np
>>> a = np.array([[2, 7, 12, 0], [3, 9, 3, 4], [4, 0, 1,  3]])
>>> a
array([[ 2,  7, 12,  0],
[ 3,  9,  3,  4],
[ 4,  0,  1,  3]])


What is the array shape?

>>> a.shape
(3, 4)


What is the array ndim?

>>> a.ndim
2


How about the len of the array?

>>> len(a)
3


Can you get the ndim and len from the shape?

>>> len(a.shape) == a.ndim
True
>>> a.shape[0] == len(a)
True


## Creating arrays using functions¶

10 minutes.

1. Create a 1D array from 2 through 5 inclusive.
>>> np.arange(2, 6)
array([2, 3, 4, 5])

1. Make an array with 10 elements between 2 and 5 inclusive.
>>> np.linspace(2, 5, 10)
array([ 2.      ,  2.333333,  2.666667,  3.      ,  3.333333,  3.666667,
4.      ,  4.333333,  4.666667,  5.      ])

1. Make an all-ones array shape (4, 4).
>>> np.ones((4, 4))
array([[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.]])

1. Make an identity array shape (6, 6).
>>> np.eye(6)
array([[ 1.,  0.,  0.,  0.,  0.,  0.],
[ 0.,  1.,  0.,  0.,  0.,  0.],
[ 0.,  0.,  1.,  0.,  0.,  0.],
[ 0.,  0.,  0.,  1.,  0.,  0.],
[ 0.,  0.,  0.,  0.,  1.,  0.],
[ 0.,  0.,  0.,  0.,  0.,  1.]])

1. Make this array with a single Python / numpy command:

1  0  0
0  2  0
0  0  3

>>> np.diag([1, 2, 3])
array([[1, 0, 0],
[0, 2, 0],
[0, 0, 3]])


Look at the docstring for np.random.randn. Make a shape (3, 5) array with random numbers from a mean=0, variance = 1 normal distribution.

>>> rand_arr = np.random.rand(3, 5)
>>> rand_arr.shape
(3, 5)


## Simple visualizations¶

7 minutes.

1. Make an array x with 100 evenly spaced values between 0 and 2 * pi;
>>> x = np.linspace(0, 2 * np.pi, 100)
>>> x.shape
(100,)

1. Make an array y which contains the cosine of the corresponding value in x - so y[i] = cos(x[i]) (hint: np.lookfor('cosine')).
>>> y = np.cos(x)
>>> y.shape
(100,)

1. Plot x against y;
>>> import matplotlib.pyplot as plt
>>> plt.plot(x, y)
[<...matplotlib.lines.Line2D object at ...>]

1. Make a 10 by 20 array of mean 0 variance 1 normal random numbers;
>>> rand_arr = np.random.randn(10, 20)
>>> rand_arr.shape
(10, 20)

1. Display this array as an image;
>>> plt.imshow(rand_arr)
<...matplotlib.image.AxesImage object at ...>

1. Investigate plt.cm. See if you can work out how to make the displayed image be grayscale instead of color.
>>> plt.imshow(rand_arr, cmap=plt.cm.gray)
<...matplotlib.image.AxesImage object at ...>


## Indexing and slicing, array creation¶

10 minutes.

1. Create the following array, call this a (you did this before):

2  7 12  0
3  9  3  4
4  0  1  3

>>> a = np.array([[2, 7, 12, 0], [3, 9, 3, 4], [4, 0, 1,  3]])

1. Get the 2nd row of a ([ 3 9 3 4]);
>>> a[1]
array([3, 9, 3, 4])

1. Get the 3rd column of a ([12 3 1]);
>>> a[:, 2]
array([12,  3,  1])

1. Create the following arrays (with correct data types):

[[1, 1, 1, 1],
[1, 1, 1, 1],
[1, 1, 1, 2],
[1, 6, 1, 1]]

[[0., 0., 0., 0., 0.],
[2., 0., 0., 0., 0.],
[0., 3., 0., 0., 0.],
[0., 0., 4., 0., 0.],
[0., 0., 0., 5., 0.],
[0., 0., 0., 0., 6.]]


Par on course: 3 statements for each

Hint: Individual array elements can be accessed similarly to a list, e.g. a[1] or a[1, 2].

Hint: Examine the docstring for diag.

>>> arr1 = np.ones((4, 4), dtype=np.int64)  # Would be float by default
>>> arr1[3, 1] = 6
>>> arr1[2, 3] = 2
>>> arr1
array([[1, 1, 1, 1],
[1, 1, 1, 1],
[1, 1, 1, 2],
[1, 6, 1, 1]])

>>> arr2 = np.diag([2., 3., 4, 5, 6], -1)  # Need a float input to diag for float output
>>> arr2[:, :-1]
array([[ 0.,  0.,  0.,  0.,  0.],
[ 2.,  0.,  0.,  0.,  0.],
[ 0.,  3.,  0.,  0.,  0.],
[ 0.,  0.,  4.,  0.,  0.],
[ 0.,  0.,  0.,  5.,  0.],
[ 0.,  0.,  0.,  0.,  6.]])

1. Skim through the documentation for np.tile, and use this function to construct the array:

[[4, 3, 4, 3, 4, 3],
[2, 1, 2, 1, 2, 1],
[4, 3, 4, 3, 4, 3],
[2, 1, 2, 1, 2, 1]]

>>> np.tile([[4, 3], [2, 1]], (2, 3))
array([[4, 3, 4, 3, 4, 3],
[2, 1, 2, 1, 2, 1],
[4, 3, 4, 3, 4, 3],
[2, 1, 2, 1, 2, 1]])


## Fancy indexing using boolean arrays¶

5 minutes.

1. Create the following array a (same as before):

2  7 12  0
3  9  3  4
4  0  1  3

>>> a = np.array([[2, 7, 12, 0], [3, 9, 3, 4], [4, 0, 1,  3]])

1. Use > to make a mask that is true where the elements are greater than 5, like this:

False True  True  False
False True  False False
False False False False

>>> mask = a > 5
array([[False,  True,  True, False],
[False,  True, False, False],
[False, False, False, False]], dtype=bool)

1. Return all the elements in a that are greater than 5.
>>> a[mask]
array([ 7, 12,  9])

1. Set all the elements greater than 5 to be equal to 5, to get this:

2  5  5  0
3  5  3  4
4  0  1  3

>>> a[mask] = 5
>>> a
array([[2, 5, 5, 0],
[3, 5, 3, 4],
[4, 0, 1, 3]])


## Elementwise operations¶

10 minutes.

Remember our array a:

2  7 12  0
3  9  3  4
4  0  1  3

1. Use array slicing to get a new array composed of the even columns (0, 2) of a. Now get array that contains the odd columns (1, 3) of a. Add these two arrays.
>>> a = np.array([[2, 7, 12, 0], [3, 9, 3, 4], [4, 0, 1,  3]])
>>> even_columns = a[:, ::2]
>>> odd_columns = a[:, 1::2]
>>> even_columns + odd_columns
array([[ 9, 12],
[12,  7],
[ 4,  4]])

1. Generate this array:

[2**0, 2**1, 2**2, 2**3, 2**4]

>>> 2 ** np.arange(5)
array([ 1,  2,  4,  8, 16])

1. Generate an array length 10 such that this is true of the elements (where x[i] is the element of x at index i):

x[i] = 2 ** (3 * i) - i

>>> inds = np.arange(10)
>>> x = 2 ** (3 * inds) - inds
>>> x
array([        1,         7,        62,       509,      4092,     32763,
262138,   2097145,  16777208, 134217719])


## Summary functions¶

Remember our array a:

2  7 12  0
3  9  3  4
4  0  1  3

>>> a = np.array([[2, 7, 12, 0], [3, 9, 3, 4], [4, 0, 1,  3]])


What are the:

• sum of all the values?
>>> a.sum()
48

• sum of the columns?
>>> a.sum(axis=0)  # Sum over the first axis, leaving the second
array([ 9, 16, 16,  7])

• sum of the rows?
>>> a.sum(axis=1)  # Sum over the second axis, leaving the first
array([21, 19,  8])

• mean?
>>> a.mean()
4.0

• min?
>>> a.min()
0

• max?
>>> a.max()
12