Exercise Numpy/Scipy

1. Basics

1. generate a 1-vector of size 10
2. generate a zero-matrix of size 4x3
3. generate a 3x3 identity matrix
4. generate 100 evenly spaced values between 7 and 9
5. Form the 2-D array (without typing it in explicitly):
   [[1, 6, 11],
   [2, 7, 12],
   [3, 8, 13],
   [4, 9, 14],
   [5, 10, 15]]
   
   
6. contruct a vector of length 100 with alternating ones and zeros ( hint: use slice indexing )
7. construct a 8x8 matrix with a checkboard pattern of zeros and ones
8. compute the inverse, eigenvectors and eigenvalues of the following matrix: $$\mathbf{A} = \left[\begin{array} {rrr} 6 & 4 & 9 \\ 1 & 0 & 2 \\ 3 & 1 & 1 \end{array}\right] $$

2. Mandelbrot

The mandelbrot set is defined as the set of complex numbers $c$ for which the iteration $z_{n+1} = (z_n)^2 + c$, with $z_0=0$ does not diverge. In this exercise we want to compute the Mandelbrot set within a certain range.

1. implement the function in_mandelbrot(real, comp) which computes whether a complex number (real = real part, comp=complex part) belongs to the mandelbrot set. Hint: the complex number 5+3i can be constructed in python as follows: c = 5 + 1j*3. Iterate N_max times and use the threshold thresh.
2. generate a 2D array with 1000x1000 zeros called M
3. compute for for all complex numbers with real part between -2 and 1 and with complex part between -1.5 and 1.5 whether they belong the Mandelbrot set and store the result in M.