*

multiplication operation: 2*3 returns 6

**

power operation: 2**3 returns 8

@

matrix multiplication:

returns

l=[a1,a2,…,an]

Constructs a list containing the objects (a1, a2,..., an). You can append to the list using l.append().The (ith) element of (l) can be accessed using l[i]

t=(a1,a2,…,an)

Constructs a tuple containing the objects (a1, a2,..., an). The (ith) element of (t) can be accessed using t[i]

len(iterable)

len is a function that takes an iterable, such as a list, tuple or numpy array and returns the number of items in that object.For a numpy array, len returns the length of the outermost dimension

returns 5.

zip

Make an iterator that aggregates elements from each of the iterables.

returns [(1,4),(2,5),(3,6)]

forainiterable:

For loop used to perform a sequence of commands (denoted using tabs) for each element in an iterable object such as a list, tuple, or numpy array.An example code is

prints [1,4,9]

ifcondition:

Performs code if a condition is met (using tabs). For example

squares (x) if (x) is (5), otherwise cubes it.

lambda

Used for create anonymous one line functions of the form:

The code after the lambda but before variables specifies the parameters. The code after the colon tells python what object to return.

def

The def command is used to create functions of more than one line:

The code immediately following def names the function, in this example g .The variables in the parenthesis are the parameters of the function. The remaining lines of the function are denoted by tab indents.The return statement specifies the object to be returned.

np.array(object,dtype=None)

np.array constructs a numpy array from an object, such as a list or a list of lists.dtype allows you to specify the type of object the array is holding.You will generally note need to specify the dtype.Examples:

A[i1,i2,…,in]

Access a the element in numpy array A in with index i1 in dimension 1, i2 in dimension 2, etc.Can use : to access a range of indices, where imin:imax represents all (i) such that (imin leq i < imax).Always returns an object of minimal dimension.For example,

A[:,2]

returns the 2nd column (counting from 0) of A as a 1 dimensional array and

A[0:2,:]

returns the 0th and 1st rows in a 2 dimensional array.

np.zeros(shape)

Constructs numpy array of shape shape. Here shape is an integer of sequence of integers. Such as 3, (1, 2), (2, 1), or (5, 5). Thus

np.zeros((5,5))

Constructs an (5times 5) array while

np.zeros(5,5)

will throw an error.

np.ones(shape)

Same as np.zeros but produces an array of ones

np.linspace(a,b,n)

Returns a numpy array with (n) linearly spaced points between (a) and (b). For example

np.linspace(1,2,10)

returns

np.eye(N)

Constructs the identity matrix of size (N). For example

np.eye(3)

returns the (3times 3) identity matrix:

np.diag(a)

np.diag has 2 uses. First if a is a 2 dimensional array then np.diag returns the principle diagonal of the matrix.Thus

np.diag([[1,3],[5,6]])

returns [1,6].

If (a) is a 1 dimensional array then np.diag constructs an array with $a$ as the principle diagonal. Thus,

np.diag([1,2])

returns

np.random.rand(d0,d1,…,dn)

Constructs a numpy array of shape (d0,d1,…,dn) filled with random numbers drawn from a uniform distribution between :math`(0, 1)`.For example, np.random.rand(2,3) returns

np.random.randn(d0,d1,…,dn)

Same as np.random.rand(d0,d1,…,dn) except that it draws from the standard normal distribution (mathcal N(0, 1))rather than the uniform distribution.

A.T

Reverses the dimensions of an array (transpose).For example,if (x = left(begin{matrix} 1& 23&4end{matrix}right)) then x.T returns (left(begin{matrix} 1& 32&4end{matrix}right))

np.hstack(tuple)

Take a sequence of arrays and stack them horizontally to make a single array. For example

returns [1,2,3,2,3,4] while

returns (left( begin{matrix} 1&22&3 3&4 end{matrix}right))

np.vstack(tuple)

Like np.hstack. Takes a sequence of arrays and stack them vertically to make a single array. For example

returns

np.amax(a,axis=None)

By default np.amax(a) finds the maximum of all elements in the array (a).Can specify maximization along a particular dimension with axis.If

a=np.array([[2,1],[3,4]])#createsa2dimarray

then

np.amax(a,axis=0)#maximizationalongrow(dim0)

returns array([3,4]) and

np.amax(a,axis=1)#maximizationalongcolumn(dim1)

returns array([2,4])

np.amin(a,axis=None)

Same as np.amax except returns minimum element.

np.argmax(a,axis=None)

Performs similar function to np.amax except returns index of maximal element.By default gives index of flattened array, otherwise can use axis to specify dimension.From the example for np.amax

returns array([1,1]) and

returns array([0,1])

np.argmin(a,axis=None)

Same as np.argmax except finds minimal index.

np.dot(a,b) or a.dot(b)

Returns an array equal to the dot product of (a) and (b).For this operation to work the innermost dimension of (a) must be equal to the outermost dimension of (b).If (a) is a ((3, 2)) array and (b) is a ((2)) array then np.dot(a,b) is valid.If (b) is a ((1, 2)) array then the operation will return an error.

np.linalg.inv(A)

For a 2-dimensional array (A). np.linalg.inv returns the inverse of (A).For example, for a ((2, 2)) array (A)

returns

np.linalg.eig(A)

Returns a 1-dimensional array with all the eigenvalues of $A$ as well as a 2-dimensional array with the eigenvectors as columns.For example,

eigvals,eigvecs=np.linalg.eig(A)

returns the eigenvalues in eigvals and the eigenvectors in eigvecs.eigvecs[:,i] is the eigenvector of (A) with eigenvalue of eigval[i].

np.linalg.solve(A,b)

Constructs array (x) such that A.dot(x) is equal to (b). Theoretically should give the same answer as

but numerically more stable.

pd.Series()

Constructs a Pandas Series Object from some specified data and/or index

pd.DataFrame()

Constructs a Pandas DataFrame object from some specified data and/or index, column names etc.

or alternatively,

plt.plot(x,y,s=None)

The plot command is included in matplotlib.pyplot.The plot command is used to plot (x) versus (y) where (x) and (y) are iterables of the same length.By default the plot command draws a line, using the (s) argument you can specify type of line and color.For example ‘-‘, ‘- -‘, ‘:’, ‘o’, ‘x’, and ‘-o’ reprent line, dashed line, dotted line, circles, x’s, and circle with line through it respectively.Color can be changed by appending ‘b’, ‘k’, ‘g’ or ‘r’, to get a blue, black, green or red plot respectively.For example,

plots the cosine function on the domain (0, 10) with a green line with circles at the points (x, v)