Vectors and Matrices

Exercise 1

Consider the following vectors in $ \mathbb{R}^7 $:

$$ u = (0.5, 0.4, 0.4, 0.5, 0.1, 0.4, 0.1), \quad v = (-1, -2, 1, -2, 3, 1, -5) $$
  1. Check if $ u $ and $ v $ are unit vectors.
  2. Calculate the dot product of the vectors $ u $ and $ v $.
  3. Are $ u $ and $ v $ orthogonal?

Exercise 2

Consider the following vectors in $ \mathbb{R}^9 $:

$$ u = (1, 2, 5, 2, -3, 1, 2, 6, 2), $$

$$ v = (-4, 3, -2, 2, 1, -3, 4, 1, -2) $$

$$ w = (3, 3, -3, -1, 6, -1, 2, -5, -7) $$
  1. Which pairs of these vectors are orthogonal?
  2. Calculate the Euclidean norm of $ u $.
  3. Calculate the infinity norm of $ w $.

Exercise 3

Consider the following matrices:

$$ A = \begin{pmatrix} 2 & -2 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & 1 \\ 6 & 2 \end{pmatrix}, \quad C = \begin{pmatrix} 4 & 1 & -1 \\ 2 & 5 & -2 \\ 1 & 1 & 2 \end{pmatrix}, \quad D = \begin{pmatrix} -3 & 1 & -1 \\ -7 & 5 & -1 \\ -6 & 6 & -2 \end{pmatrix} $$

$$ E = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}, \quad F = \begin{pmatrix} -2 & 1 & 0 \end{pmatrix}, \quad G = \begin{pmatrix} 1 & -1 & 0 & 0 \\ 1 & 4 \end{pmatrix} $$

  1. Calculate, if possible:

    • $ A + B $
    • $ B - A $
    • $ B + C $
    • $ AB $
    • $ BA $
    • $ BG $
    • $ CE $
    • $ EF $
    • $ FE $

  2. Write the transposes of $ A $ and $ B $ and calculate their product. Which property can one observe?

Exercise 4

Consider the following matrices:

$$ A = \begin{pmatrix} 2 & -2 \\ -3 & 1 \\ 5 & -3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 4 & 4 \\ -2 & 3 & -7 \\ 2 & 5 & -7 \end{pmatrix}, \quad C = \begin{pmatrix} 4 & -1 & 2 \\ -8 & 2 & -4 \\ 2 & 1 & -4 \end{pmatrix} $$
  1. Compute $ A^T B $ and $ C + B $.
  2. Which of the matrices $ A $, $ B $, $ C $ are full rank?
  3. Calculate the Frobenius norm of $ C $ and the spectral norm of $ A $.
  4. Calculate the inverse of $ B $.

Exercise 5

Consider the matrices of Exercise 3.

  1. Calculate the determinants of the matrices $ A $, $ B $, and $ AB $.
  2. Calculate the determinants of the matrices $ C $ and $ D $.

Exercise 6

Consider the following matrices:

$$ A = \begin{pmatrix} 2 & -1 \\ 4 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 2 & 0 \\ 4 & 5 \end{pmatrix}, \quad C = \begin{pmatrix} 6 & -9 \\ -4 & 6 \end{pmatrix}, \quad D = \begin{pmatrix} -1 & 6 & 2 \\ 0 & 1 & 0 \\ 3 & 0 & -5 \end{pmatrix} $$

Calculate, if possible, the inverses of the matrices $ A $, $ B $, $ C $, and $ D $.

Exercise 7

Consider the matrix:

$$ A = \begin{pmatrix} 2 & 2 & 3 \\ -2 & 7 & 4 \\ -3 & -3 & -4 \\ -8 & 2 & 3 \end{pmatrix} $$
  1. Add a column to $ A $ so that it is invertible.
  2. Remove a row from $ A $ so that it is invertible.
  3. Calculate $ AA^T $. Is it invertible?
  4. Calculate $ A^T A $. Is it invertible?

Exercise 8

  1. Calculate the inverse of the matrix $ M = \begin{pmatrix} 3 & 2 & -1 \ 1 & -1 & 1 \ 2 & -4 & 5 \end{pmatrix} $.
  2. 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

Solve the systems:

$$ \begin{cases} 2x + 3y + 5z = 2 \\ 7x + z = -1 \\ -2y + 2z = 3 \end{cases} $$
$$ \begin{cases} x + 2y - z = 2 \\ 2x + 5y + 4z = 3 \\ 3x + 7y + 4z = 1 \end{cases} $$