To determine a matrix's eigenvalues and eigenvectors in Python, use the eig function from the numpy linalg library. As shown in the code below, it gives the eigenvalues and eigenvectors as one eigenvector per column of a matrix. The eigenvalues of the matrix A are contained in the vector w, and the matrix V has one eigenvector for each column. The determinant of the given matrix A is equal to the product of its Eigenvalues, which are 2, 1, and 4, and the trace of A is equivalent to the sum of those values.

The following equation can be used to compute values of y as a function of x: y=be¯ªsin(bx) (0.0012x¹ −0.15x³ +0.075x² +2.5x) where a and b are parameters. (a) Write Python code to compute a vector y for a = 2, b = 5, and x as a vector with values from 0 to π/2 in increments Ax= π/40. (b) Compute the vector z = y² with element-by-element-wise operations. (c) Combine x, y, and zinto a 3-column matrix W.(Hint use np. hstack function) (d) Generate a plot of y and z versus x. Include a title, axis labels, and a legend. For the y versus x curve, format a solid blue line. For the z versus x curve, choose a magenta dashed line. Include grid lines.

The following equation can be used to compute values of y as a function of x: y=be¯ªsin(bx) (0.0012x¹ −0.15x³ +0.075x² +2.5x) where a and b are parameters. (a) Write Python code to compute a vector y for a = 2, b = 5, and x as a vector with values from 0 to π/2 in increments Ax= π/40. (b) Compute the vector z = y² with element-by-element-wise operations. (c) Combine x, y, and zinto a 3-column matrix W.(Hint use np. hstack function) (d) Generate a plot of y and z versus x. Include a title, axis labels, and a legend. For the y versus x curve, format a solid blue line. For the z versus x curve, choose a magenta dashed line. Include grid lines.

3. Use the above functions to write a function EULERXYZ ( ) which accepts one input - a 3-vector of angles (in radians), and returns the corresponding 3 × 3 rotation matrix. Experiment plotting various general rotations.