Show that in space $\mathbb R^3 $ the vectors $x=(1,1,0), y=(0,1,2) $ and $z=(3,1,-4)$ are linearly independent

Question

I am having a hard time in proving this.

It would be a great help.

Can someone help me with this?

***without the matrix form

Answer 1

Hint: Solve the System $$\alpha[1,1,0]+\beta[0,1,2]+\gamma[3,1,-4]=[0,0,0]$$

Answer 2

They're not linearly independent: $z=3x-2y$

Answer 3

If they are linearly independent, their scalar product should be non-zero.

$$[\vec a,\vec b, \vec c] = \begin{vmatrix} 1 & 1 & 0 \\ 0 & 1 & 2 \\ 3 & 1 & -4 \end{vmatrix} = 0. $$ So the given vectors are linearly dependent.