Question

Care din urmatoarele sisteme de vectori din spatiul vectorial aritmetic R³ sunt liniar independente: a)(₁ = (1,2,3), z = (2,3,1), 3 = (3,1,2)} b) b = (1,3,-1), (0, -2, 1), 53 = (-3,-1,-1)}​

Answer 1

$\vec{a_1}=(1,2,3)\\\\\vec{a_2}=(2,3,1)\\\\\vec{a_3}=(3,1,2)$

Fie combinatia liniara:

$\alpha_1\vec{a_1}+\alpha_2\vec{a_2}+\alpha_3\vec{a_3}=\vec{0}$

⇒ $\alpha_1(1,2,3)+\alpha_2(2,3,1)+\alpha_3(3,1,2)=\vec{0}$

⇒ $(\alpha_1,2\alpha_1,3\alpha_1)+(2\alpha_2,3\alpha_2,\alpha_2)+(3\alpha_3,\alpha_3,2\alpha_3)=\vec{0}$

⇒ $(\alpha_1+2\alpha_2+3\alpha_3,2\alpha_1+3\alpha_2+\alpha_3,3\alpha_1+\alpha_2+2\alpha_3)=\vec{0}$

Vom rezolva sistemul:

$\alpha_1+2\alpha_2+3\alpha_3=0\\\\2\alpha_1+3\alpha_2+\alpha_3=0\\\\3\alpha_1+\alpha_2+2\alpha_3=0$

Avem matricea sistemului:

$\Delta=\left|\begin{array}{ccc}1&2&3\\2&3&1\\3&1&2\end{array}\right|$ =(6+6+6)-(27+1+8)=-18

           1   2   3

           2  3   1

sistemul admite solutia unica 0⇒sistemul de vectori este liniar independent

Mai multe despre vectori liniari gasesti aici: https://brainly.ro/tema/4705502

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