Question

A set of vectors $\{(a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3)\}$ is said to be linearly independent if and only if
$\begin{vmatrix}
a_{1} & a_{2} & a_{3}\\
b_{1} & b_{2} & b_{3}\\
c_{1} & c_{2} & c_{3}
\end{vmatrix}\neq 0$
otherwise the set is said to be linearly dependent. A similar result holds for $\{(a_1,a_2),(b_1,b_2)\}$.

If $(a_1,a_2,a_3)$, $(b_1,b_2,b_3)$ and $(c_1,c_2,c_3)$ are linearly independent and $x,y,zϵR$, then

• A. there exist $\alpha ,\beta ,\gamma ϵR$ such that $(x,y,z)+\alpha (a_1,a_2,a_3)+\beta (b_1,b_2,b_3)+\gamma (c_1,c_2,c_3)=(0,0,0)$
• B. if $x(a_1,a_2,a_3)+y(b_1,b_2,b_3)+z(c_1,c_2,c_3)=(0,0,0)\Rightarrow x+y+z\ne 0$.
• C. $\alpha (a_1,a_2,a_3)+\beta (b_1,b_2,b_3)+\gamma (c_1,c_2,c_3)=(x,y,z)$ cannot hold for any values of $\alpha ,\beta ,\gamma$
• D. none of these

Answer 1

The correct option is A there exist $\alpha ,\beta ,\gamma ϵR$ such that $(x,y,z)+\alpha (a_1,a_2,a_3)+\beta (b_1,b_2,b_3)+\gamma (c_1,c_2,c_3)=(0,0,0)$

$(a_1,a_2,a_3),(b_1,b_2,b_3)$ and

$(c_1,c_2,c_3)$ are linearly independent when

$a_1\alpha +b_1\beta +c_1\gamma =-x{a}_2\alpha +b_2\beta +c_2\gamma =-y{a}_3\alpha +b_3\beta +c_3\gamma =-z$

From Cramer's rule

$\alpha (a_1,a_2,a_3)+\beta (b_1,b_2,b_3)+\gamma (c_1,c_2,c_3)=-(x,y,z)$

$\Rightarrow (x,y,z)+\alpha (a_1,a_2,a_3)+\beta (b_1,b_2,b_3)+\gamma (c_1,c_2,c_3)=(0,0,0)$