Question

Escreva o vetor u = 2i – 3j como combinação linear dos vetores v1 = -2i -4j e v2 = 3i – j.

Answer 1

Olá, Eric.

 

Devemos encontrar  $<var>\lambda_1,\lambda_2</var>$  tais que:

 

$<var>u=\lambda_1v_1+\lambda_2v_2 \Rightarrow \\\\ 2i-3j=\lambda_1 (-2i -4j)+\lambda_2( 3i-j) \Rightarrow \\\\ 2i-3j=-2\lambda_1i -4\lambda_1j+3\lambda_2i-\lambda_2j \Rightarrow \\\\ 2i-3j=(-2\lambda_1+3\lambda_2)i +(-4\lambda_1-\lambda_2)j \Rightarrow \\\\ \begin{cases} -2\lambda_1+3\lambda_2=2\\ -4\lambda_1-\lambda_2=-3\ \ (\times 3)\end{cases} \Rightarrow \begin{cases} -2\lambda_1+3\lambda_2=2\\ -12\lambda_1-3\lambda_2=-9\end{cases}\ \bigoplus\Rightarrow$

 

$-14\lambda_1=-7 \Rightarrow \boxed{\lambda_1=\frac12} \Rightarrow </var>$

 

$<var>-1+3\lambda_2=2 \Rightarrow \boxed{\lambda_2=1}</var>$

 

$<var>\therefore u=\lambda_1v_1+\lambda_2v_2 \Rightarrow \boxed{u=\frac12 v_1+v_2}</var>$