Exercises

Ex. 1 → Consider a 4 × 3 matrix M:

1. Construct this matrix in Python using the function array .
2. Construct the same matrix using the function arange  followed by a suitable reshape.
3. What is the result of the expression M[2,:] ? What is the result of the similar expression M[2:]?

Ex. 2 → Given a vector x, construct in Python the following matrix:

                 

Here, x_i are the components of the vector x (numbered from zero). Given a vector y, solve in Python the linear equation system Va = y. Let the components of a be denoted by a_i, i = 0, ..., 5. Write a function poly, which has a and z as input and which computes the polynomial:

Plot this polynomial and depict in the same plot the points (x_i , y_i ) as small stars. Try your code with the vectors:

• x = (0.0, 0.5, 1.o, 1.5, 2.0, 2.5)
• y = (-2.0, 0.5, -2.0, 1.0, -0.5, 1.0)

Ex. 3 → The matrix V in Ex. 2 is called a Vandermonde matrix. It can be set up in Python directly by the command vander. Evaluating a polynomial defined by a coefficient...