Homework: Vectors and matrices

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  • Python-scripts are submitted in one file called: homework3.py

  • Please do NOT change the name of this script file.

This homework requires you to manipulate vectors and matrices. It makes extensive use of the numpy (Numerical Python) library. Don’t forget to import it first.

Exercise 1

Define the following matrices as A and B

\[\begin{split}\left( \begin{array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array} \right)\end{split}\]
\[\begin{split}\left( \begin{array}{ccc} 0 & 2 & 3 \\ 0 & 5 & 0 \\ 7 & 0 & 0 \end{array} \right)\end{split}\]

  1. Multiply A and B element by element.

  2. Add A and B element by element.

  3. Transpose matrix A to A’.

  4. Extract the second row of matrix A, that is 1 0 1 and print it.

  5. Use matrix slicing to define vector a as the last column of matrix A and vector b as the last row of matrix B. Do not simply hard code the numbers of these vectors but use slicing, like you did in the previous question. Then add up a and b and print the result.

Exercise 2

The aim is to fill two vector arrays xv and yv with x and f(x) values, respectively, where

\(f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{- \frac{1}{2} \left( \frac{x- \mu}{\sigma} \right) ^2}\)

is the density of the standard normal distribution, that is a normal density with mean \(\mu=0\) and standard deviation \(\sigma = 1\). Let the xv values be uniformly spaced in xv = [-4, 4] with 100 grid points. Write a loop to fill in the values for \(xv\) and \(yv=f(xv)\). So at the end you should have a vector xv with the X-values and a vector yv with the function values \(f(x)\).

Hint: The constant pi, the square root function \(\sqrt{x}\) (sqrt()) and the natural exponential function \(e\) (exp()) are all coded in the numpy library. No need to use the math library for these.

Bonus points: Plot yv against xv using the plot command.

Exercise 3

Create a vector array wv with values 0, 0.1, 0.2, …, 3 without using a loop. Write print statements to print w[:], w[:-2], w[::5], w[2:-2:6]. Explain which elements of the array are printed using these commands!

Exercise 4

Create a \(3 \times 3\) matrix with the value \(1\) in the main diagonal and call this matrix A. Then create a \(5 \times 3\) matrix filled with the value \(1\) and call it B. When I say create I don’t mean to simply type these matrices by hand. Use some built in numpy functions that generate these specific types of matrices automatically. Read the book carefully, the answer is right there!

Then create a third matrix C that stacks the two earlier matrices on top of each other so that the result looks like this:

\[\begin{split} C = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array} \right)\end{split}\]

Exercise 5

  1. Use the matrix from Exercise 4 and run a loop that extracts all the elements from the second column and prints them one by one.

  2. Use the matrix from Exercise 4 and run a loop that extracts all the elements from the third row and prints them one by one.

Exercise 6

Create the following matrix A by hand (just type it into the script file).

\[\begin{split}\left( \begin{array}{ccc} 2 & 0 & 10 \\ 0 & 3 & -50 \\ 0 & 0 & 4 \\ 6 & 0 & 0 \\ 5 & 7 & 0 \\ 0 & 0 & 8 \end{array} \right)\end{split}\]

Write a loop (or a nested loop) that extracts the diagonal elements 2,3,4,6,7,8 and prints them one by one.