Vettori

di
Algebra lineare

\displaystyle \textbf{a} = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix}, \displaystyle \textbf{b} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}

\displaystyle \textbf{0} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}

\displaystyle \textbf{i} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \displaystyle \textbf{j} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \displaystyle \textbf{k} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}

Prodotto per uno scalare

\displaystyle \textbf{c} = k\cdot \textbf{a} = \displaystyle k\cdot \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} = \displaystyle\begin{bmatrix} k\cdot a_1 \\ k\cdot a_2 \\ k\cdot a_3 \end{bmatrix}

Proprietà

  • \displaystyle c_i = k\cdot a_i
  • \displaystyle k\cdot \textbf{a} = \textbf{a}\cdot k
  • \displaystyle 0\cdot \textbf{a} = \textbf{0}

Somma

\displaystyle \textbf{c} = \textbf{a}+\textbf{b} = \displaystyle \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} + \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} = \displaystyle \begin{bmatrix} a_1+b_1 \\ a_2+b_2 \\ a_3+b_3 \end{bmatrix}

Proprietà

  • \displaystyle c_i = a_i+b_i
  • \displaystyle \textbf{a}+\textbf{b} = \textbf{b}+\textbf{a}
  • \displaystyle (\textbf{a}+\textbf{b})+\textbf{c} = \textbf{a}+(\textbf{b}+\textbf{c})

Opposto

\displaystyle \textbf{c} = -\textbf{a} = \displaystyle (-1)\cdot \textbf{a} = \displaystyle (-1)\cdot \begin{bmatrix} a_1 \\ a_2 \\ a_3\end{bmatrix} = \displaystyle \begin{bmatrix} -a_1 \\ -a_2 \\ -a_3\end{bmatrix}

Proprietà

  • \displaystyle c_i = -a_i
  • \displaystyle \textbf{a}+(-\textbf{a}) = \textbf{0}
  • \displaystyle -\textbf{0} = \textbf{0}

Sottrazione

\displaystyle \textbf{c} = \textbf{a}-\textbf{b} = \displaystyle \begin{bmatrix} a_1-b_1 \\ a_2-b_2 \\ a_3-b_3\end{bmatrix}

Proprietà

  • \displaystyle c_i = a_i-b_i
  • \displaystyle \textbf{a}-\textbf{b} = \textbf{a}+(-\textbf{b})

Prodotto scalare

\displaystyle x=\textbf{a}^T \textbf{b} = \displaystyle \begin{bmatrix} a_1 & a_2 & a_3 \end{bmatrix}\cdot \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} = \displaystyle a_1\cdot b_1+a_2\cdot b_2+a_3\cdot b_3

Proprietà

  • \displaystyle \textbf{a}^T \textbf{0} = \textbf{0}^T \textbf{a} = 0
  • \displaystyle \textbf{i}^T \textbf{j} = \textbf{i}^T \textbf{k} = \textbf{j}^T \textbf{k} = 0
  • \displaystyle \textbf{i}^T \textbf{i} = \textbf{j}^T \textbf{j} = \textbf{k}^T \textbf{k} = 1
  • …

Norma

  • \displaystyle ||\textbf{a}|| \ge 0
  • \displaystyle ||\textbf{0}|| = 0
  • \displaystyle ||\textbf{a}+\textbf{b}|| \le ||\textbf{a}|| + ||\textbf{b}||
  • \displaystyle ||k\cdot \textbf{a}|| = |k|\cdot ||\textbf{a}||

Norma 1

\displaystyle ||\textbf{a}||_1 = \sum_{i=1}^{n} |x_i|

= |a_1| + |a_2| + \dots + |a_n|

La somma dei valori assoluti

Norma 2

Norma euclidea
Lunghezza

\displaystyle ||\textbf{a}||_2
= \sqrt{a_1^2 + a_2^2 + \dots + a_n^2}

  • \displaystyle ||\textbf{a}|| = \sqrt{\textbf{a}^T \textbf{a}}
  • \displaystyle |\textbf{a}\cdot \textbf{b}| \le ||\textbf{a}||\cdot ||\textbf{b}||
  • \displaystyle \textbf{a}\cdot \textbf{b} = ||\textbf{a}||\cdot ||\textbf{b}||\cdot \cos(\theta)
  • …

Norma 3

Norma infinito

\displaystyle ||\textbf{a}||_3 = ||\textbf{a}||_{\infty}
= \max \{|x_1|, |x_2|, ..., |x_n|\}

Il valore assoluto maggiore