Matrices and vectors - Your turn

Matrices and vectors - Your turn#

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Part (a)#

Find the dot product between the first and second rows of matrix \( C = \begin{bmatrix} 5 & 2 & 1 \\ 5 & 3 & -1 \end{bmatrix}\).

# TODO: Write your solution below

Part (b)#

Compute \(B^T A^T - B^T\) using the matrices \(A = \begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}\) and \(B = \begin{bmatrix} 1 & 0 & 1 \\ 2 & 1 & 0 \end{bmatrix}\).

# TODO: Write your solution below

Part (c)#

A rectangle is formed by the four vertices: \((1,0)\), \((3,0)\), \((3,2)\), and \((1,2)\). Compute the coordinates of each vertex after a rotation of \(60^{\circ}\) (or \(\frac{\pi}{3}\) rad) using a rotation matrix \(R\). Plot the rectangle before and after the rotation.

Hint 1: For a counter-clockwise rotation from point \(A\,(x_A, y_A)\) to \(B\,(x_B, y_B)\) about the origin through an angle \(\theta\), it is given by:

\[\begin{split} \begin{bmatrix} x_B \\ y_B \end{bmatrix} = R \begin{bmatrix} x_A \\ y_A \end{bmatrix} \quad \text{with} \quad R = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \end{split}\]

Hint 2: To plot any set of points linked together in Python, create a matrix whose first row contains the \(x\)-components of each point and the second row the \(y\)-components. Then plot the first line versus the second one. To close your rectangles, use the first vertex as first and last point.

Hint 3: You may want to add the line ax.set(aspect='equal') for your axes, just so you get the rectangle you expect!

# TODO: Write your solution below

Part (d)#

Each year, a college accepts \(1000\) freshmen and \(200\) sophomores into the school. Also, \(10\%\) of the freshmen, \(25\%\) of the sophomores, \(5\%\) of the juniors, and \(5\%\) of the seniors repeat their year. The rest continue their studies. If there are \(1160\) freshmen, \(1057\) sophomores, \(1183\) juniors, and \(1028\) seniors this year and the trend holds, what is the school population in \(10\) years? The system of equations to use is as follows:

\[\begin{split} \begin{align*} x_1(t+1) &= 0.1 x_1(t) + 1000 \\ x_2(t+1) &= 0.9 x_1(t) + 0.25 x_2(t) + 200 \\ x_3(t+1) &= 0.75 x_2(t) + 0.05 x_3(t) \\ x_4(t+1) &= 0.95 x_3(t) + 0.05 x_4(t) \end{align*} \end{split}\]

where the number is indicative of the student’s year in college (\(1\) = freshmen, \(2\) = sophomore, etc). From the equations, we can derive an equation \(x^{(t+1)} = A x^{(t)} + b\) with

\[\begin{split} A = \begin{bmatrix} 0.1 & 0 & 0 & 0 \\ 0.9 & 0.25 & 0 & 0 \\ 0 & 0.75 & 0.05 & 0 \\ 0 & 0 & 0.95 & 0.05 \end{bmatrix}, \qquad b = \begin{bmatrix} 1000 \\ 200 \\ 0 \\ 0 \end{bmatrix}, \qquad x_0 = \begin{bmatrix} 1160 \\ 1057 \\ 1183 \\ 1028 \end{bmatrix} \end{split}\]

Hint: When displaying the final population vector, use the np.round() function to display integral values. We cannot have partial students.

# TODO: Write your solution below

Exporting your work#

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