Vectors and Matrices in Python
In this worksheet, you will use Python (NumPy) to perform vector and matrix operations.
For each exercise, write Python code to compute the required results and verify them numerically.
Exercise 1 : Vectors in $\mathbb{R}^7$
Consider the following vectors:
$$ u = (0.5, 0.4, 0.4, 0.5, 0.1, 0.4, 0.1), \quad v = (-1, -2, 1, -2, 3, 1, -5) $$Using Python and NumPy:
- Check whether $u$ and $v$ are unit vectors.
- Compute the dot product of $u$ and $v$.
- Determine if $u$ and $v$ are orthogonal.
Exercise 2 : Norms and Orthogonality
Consider the following vectors in $\mathbb{R}^9$:
$$ u = (1, 2, 5, 2, -3, 1, 2, 6, 2), \quad v = (-4, 3, -2, 2, 1, -3, 4, 1, -2), \quad w = (3, 3, -3, -1, 6, -1, 2, -5, -7) $$Using Python:
- Check which pairs of these vectors are orthogonal.
- Calculate the Euclidean norm of $u$.
- Calculate the infinity norm of $w$.
Exercise 3 : Matrix Operations
Consider the matrices:
$$ A = \begin{pmatrix} 2 & -2 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & 1 \\ 6 & 2 \end{pmatrix}, \quad C = \begin{pmatrix} 4 & 1 & -1 \\ 2 & 5 & -2 \\ 1 & 1 & 2 \end{pmatrix}, \quad D = \begin{pmatrix} -3 & 1 & -1 \\ -7 & 5 & -1 \\ -6 & 6 & -2 \end{pmatrix} $$$$ E = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}, \quad F = \begin{pmatrix} -2 & 1 & 0 \end{pmatrix}, \quad G = \begin{pmatrix} 1 & -1 & 0 & 0 \\ 1 & 4 & 0 & 0 \end{pmatrix} $$
Using Python:
- Compute (if possible):
- $A + B$, $B - A$, $B + C$, $AB$, $BA$, $BG$, $CE$, $EF$, $FE$
- Compute the transposes of $A$ and $B$ and then their product.
Observe and explain any property you find.
Exercise 4 : Matrix Rank and Norms
Consider the following matrices:
$$ A = \begin{pmatrix} 2 & -2 \\ -3 & 1 \\ 5 & -3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 4 & 4 \\ -2 & 3 & -7 \\ 2 & 5 & -7 \end{pmatrix}, \quad C = \begin{pmatrix} 4 & -1 & 2 \\ -8 & 2 & -4 \\ 2 & 1 & -4 \end{pmatrix} $$Using Python:
- Compute $A^T B$ and $C + B$.
- Determine which of $A$, $B$, and $C$ are full rank.
- Compute the Frobenius norm of $C$ and the spectral norm of $A$.
- Attempt to compute the inverse of $B$.
Exercise 5 : Determinants
Using the matrices from Exercise 3, and Python:
- Compute $\det(A)$, $\det(B)$, and $\det(AB)$.
- Compute $\det(C)$ and $\det(D)$.
Exercise 6 : Inverses
Consider the matrices:
$$ A = \begin{pmatrix} 2 & -1 \\ 4 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 2 & 0 \\ 4 & 5 \end{pmatrix}, \quad C = \begin{pmatrix} 6 & -9 \\ -4 & 6 \end{pmatrix}, \quad D = \begin{pmatrix} -1 & 6 & 2 \\ 0 & 1 & 0 \\ 3 & 0 & -5 \end{pmatrix} $$Using Python, calculate (if possible) the inverses of $A$, $B$, $C$, and $D$.
Exercise 7 : Invertibility
Consider the matrix:
$$ A = \begin{pmatrix} 2 & 2 & 3 \\ -2 & 7 & 4 \\ -3 & -3 & -4 \\ -8 & 2 & 3 \end{pmatrix} $$Using Python:
- Add a column to $A$ to make it invertible.
- Remove a row from $A$ to make it invertible.
- Compute $AA^T$ and check if it is invertible.
- Compute $A^T A$ and check if it is invertible.
Exercise 8 : Matrix Inversion and Systems of Equations
Using Python:
- Compute the inverse of $$ M = \begin{pmatrix} 3 & 2 & -1 \\ 1 & -1 & 1 \\ 2 & -4 & 5 \end{pmatrix}. $$
- Use this inverse to solve the linear system: $$ \begin{cases} 3x + 2y - z = 5 \\ x - y + z = 1 \\ 2x - 4y + 5z = -3 \end{cases} $$
Exercise 9 : Solving Linear Systems
Use Python to solve the following systems: