Question

determine whether the vectors (1,3,-1,4),(3,8,-5,7),(2,9,4,23) in R^4 are linearly independent or linearly dependent 

Answer 1

Let us determine whether the vectors $(1,3,-1,4),\ (3,8,-5,7),\ (2,9,4,23)$ in $\R^4$ are linearly independent or linearly dependent. Let us consider the equality

$a(1,3,-1,4)+b (3,8,-5,7)+c (2,9,4,23)=(0,0,0,0).$

It follows that $(a+3b+2c, 3a+8b+9c,-a-5b+4c,4a+7b+23c)=(0,0,0,0),$ and hence we have the following system:

$\begin{cases} a+3b+2c=0\\ 3a+8b+9c=0 \\ -a-5b+4c=0 \\ 4a+7b+23c=0\end{cases}$

After adding to the second equation the first multiplied by -3, to the third equation the first, and to the fourth equation the first multiplied by -4, we conclude that the last system is equivalent to following the system:

  $\begin{cases} a+3b+2c=0\\ -b+3c=0 \\ -2b+6c=0 \\ -5b+15c=0\end{cases}$

After dividing the third equation by 2 and the fourth equation by 5, we conclude that the second equation coinside with the third equation and the fourth equation. Therefore, we have the following equivalent systems:

$\begin{cases} a+3b+2c=0\\ b=3c \\ c\in\R\end{cases}$

$\begin{cases} a=-3b-2c=-11c\\ b=3c \\ c\in\R\end{cases}$

Let $c=1,$ then $b=3,\ a=-11.$ We have that $-11(1,3,-1,4)+3(3,8,-5,7)+(2,9,4,23)=(0,0,0,0),$ and hence we conclude that this vectors are linearly dependent.