ukol_1

6. Vypočítejte souřadnice v ∈ ℝ3 v bázi F = (f1,f2,f3), kde
v = (2,−1,−1), f1 = (0, 1, 1), f2 = (−2,−2, 0), f3 = (−2,−1,−1).

$$\left( \begin{matrix} 0 & - 2 & - 2 \\ 1 & - 2 & - 1 \\ 1 & 0 & - 1 \\ \end{matrix} \middle| \begin{matrix} 2 \\ - 1 \\ - 1 \\ \end{matrix} \right)\begin{matrix} r.3 \\ - r.\ 3 \\ r.\ 1 \\ \end{matrix}\sim\left( \begin{matrix} 1 & 0 & - 1 \\ 0 & - 2 & 0 \\ 0 & - 2 & - 2 \\ \end{matrix} \middle| \begin{matrix} - 1 \\ 0 \\ 2 \\ \end{matrix} \right)\begin{matrix} \\ \\ - r.\ 2,\ /( - 2) \\ \end{matrix}\sim\left( \begin{matrix} 1 & 0 & - 1 \\ 0 & - 2 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \middle| \begin{matrix} - 1 \\ 0 \\ - 1 \\ \end{matrix} \right)\begin{matrix} + r.\ 3 \\ /( - 2) \\ \\ \end{matrix}\sim\left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \middle| \begin{matrix} - 2 \\ 0 \\ - 1 \\ \end{matrix} \right)$$

v = -2f1+0f2-f3

F = (-2,0,-1)

$$\left\lbrack V \right\rbrack_{F} = \begin{pmatrix} - 2 \\ 0 \\ - 1 \\ \end{pmatrix}$$

7. Určete bázi a dimenzi V

V = {ax2 + bx + c ∈ Ƥ2|a − c = 0 ∧ b = a}.

a + 0 − c = 0

 − a + b + 0 = 0

$$\left( \begin{matrix} 1 & 0 & - 1 \\ - 1 & 1 & 0 \\ \end{matrix} \middle| \begin{matrix} 0 \\ 0 \\ \end{matrix} \right)\begin{matrix} \\ + \ r.\ \ 1 \\ \end{matrix}\sim\left( \begin{matrix} 1 & 0 & - 1 \\ 0 & 1 & - 1 \\ \end{matrix} \middle| \begin{matrix} 0 \\ 0 \\ \end{matrix} \right)$$

α3 = t, α2 = t, α1 = t

$F = \ \begin{pmatrix} t \\ t \\ t \\ \end{pmatrix} = t\begin{pmatrix} 1 \\ 1 \\ 1 \\ \end{pmatrix}$, dim V = 1