1.
O vetor \(5 + 3x +2x^2 \in \mathcal{P}_2[x](\mathbb{R})\) é combinação linear dos vetores \(1+ x + x^2\) e \(3+ x\text{?}\)
Seção 4.7 Exercícios
Exercícios Exercícios
2.
Verifique se os seguintes conjuntos são L.I. ou L.D.
- \(S=\{(1,\,1,\,0),\,(1,\,0,\,1),\,(0,\,1,\,1)\}\subset \mathbb{R}^3\text{.}\)
- \(S=\{1+2x,\, x+ x^3,\, x^2+ x^3 \}\subset \mathcal{P}_3[x] (\mathbb{R})\text{.}\)
- \(S=\left\{ \left[\begin{array}{rr} 1 \amp 0 \\ 1 \amp 1 \end{array}\right],\, \left[\begin{array}{rr} -1 \amp 2 \\ 0 \amp 1 \end{array}\right],\, \left[\begin{array}{rr} 0 \amp -1 \\ 2 \amp 1 \end{array}\right],\, \left[\begin{array}{rr} 1 \amp 8 \\ 0 \amp 5 \end{array}\right]\right\}\text{.}\)
- \(S= \{(-1,\,1,\,0),\,(0,\,1,\,-2),\,(-2,\,3,\,1)\}\subset \mathbb{R}^3\text{.}\)
- \(S=\{(1,\,2,\,-1),\,(-1,\,1,\,0),\,(-3,\,0,\,1),\,(-2,\,-1,\,1) \}\subset \mathbb{R}^3\text{.}\)
- \(S=\{2x+2,\, -x^2 + x+3,\,x^2 + 2x\}\subset \mathcal{P}_2[x](\mathbb{R})\text{.}\)
3.
Considere \(S\) o subespaço de \(\mathcal{M}_{2\times 2}(\mathbb{R})\) descrito por
\(S =\left\{ \left[\begin{array}{rr} a-b \amp 2a \\ a+b \amp -b \end{array}\right] \;\bigg|\; a,\,b\in \mathbb{R} \right\}\text{.}\) - \(\left[\begin{array}{rr} 5 \amp 6 \\ 1 \amp 2 \end{array}\right] \in S\text{?}\)
- Encontre um valor para \(k\) de forma que o vetor \(\left[\begin{array}{rr} -4 \amp k \\ 2 \amp -3 \end{array}\right]\) pertença a \(S\text{.}\)
4.
Considere os vetores \(v_1 = (1,\,1,\,1),\,v_2 = (1,\,2,\,0)\) e \(v_3 = (1,\,3,\,-1)\text{.}\) Se \((3,\,-1,\,k) \in [v_1 ,\, v_2 ,\, v_3]\) (espaço gerado pelos vetores \(v_1,\,v_2\) e \(v_3\)), qual o valor de \(k\text{?}\)
5.
Mostre que os vetores \(v_1= (1,\,1,\,1),\,v_2 = (0,\,1,\,1)\) e \(v_3=(0,\,0,\,1)\) geram o espaço euclidiano \(\mathbb{R}^3\text{.}\)
6.
Verifique que o vetor \(w=(-1,\,-3,\,2,\,0)\) pertence ao subespaço de \(\mathbb{R}^4\) gerado pelos vetores \(v_1=(2,\,-1,\,3,\,0),\,v_2=(1,\,0,\,1,\,0)\) e \(v_3=(0,\,1,\,-1,\,0)\text{.}\)
