## klimenkov 3 years ago A tutorial for @RolyPoly. Linear independence of the vectors. Check if the vectors $$\vec{v_1}=(1,0,1), \vec{v_2}=(2,1,3), \vec{v_3}=(1,1,2)$$ are linearly independent?

1. anonymous

$\left[\begin{matrix}1 & 2 & | & 1 \\ 0 & 1 &|& 1 \\ 1&3&|&2\end{matrix}\right]$$->\left[\begin{matrix}1 & 2 & | & 1 \\ 0 & 1 &|& 1\\ 0&1&|&1\end{matrix}\right]$$->\left[\begin{matrix}1 & 2 & | & 1 \\ 0 & 1 &|& 1\\ 0&0&|&0\end{matrix}\right]$ => Linearly dependent :|

2. klimenkov

We will use a definition of the linear dependence. Start from making the linear combination of $$\vec{v_1},\vec{v_2},\vec{v_3}$$: $$c_1\cdot\vec{v_1}+c_2\cdot\vec{v_2}+c_3\cdot\vec{v_3}$$ And we will try to check if there are such $$c_1,c_2,c_3$$ to make this combination equal to zero. $$c_1\cdot\left(\begin{matrix}1\\0\\1\end{matrix}\right)+c_2\cdot\left(\begin{matrix}2\\1\\3\end{matrix}\right)+c_3\cdot\left(\begin{matrix}1\\1\\2\end{matrix}\right)=\left(\begin{matrix}0\\0\\0\end{matrix}\right)$$. That means that we have a system of the linear equation with the right part equal to zero: $$A\vec{x}=0$$ Where $$A=\left(\begin{matrix} 1&2&1\\ 0&1&1\\ 1&3&2 \end{matrix}\right), \vec{x}=\left(\begin{matrix}c_1\\c_2\\c_3\end{matrix}\right)$$. Using Cramer's rule we compute the determinant: $$\left|\begin{matrix} 1&2&1\\ 0&1&1\\ 1&3&2 \end{matrix}\right|=0$$ We have that the right side of the matrix equation $$A\vec{x}=0$$ is equal to zero and the determinant of the matrix $$A$$ is equal to zero. That means that there is non-trivial solution for $$\vec{x}$$. We found that there are so $$c_1,c_2,c_3$$ not all are equal to zero. According to the definition of the linear independence of the vector, we can say that $$\vec{v_1},\vec{v_2},\vec{v_3}$$ are linearly dependent.

3. anonymous

We have that the right side of the matrix equation $$A\vec{x} =0$$ is equal to zero and the determinant of the matrix A is equal to zero. ^ Okay. That means that there is non-trivial solution for $$\vec{x}$$ ^ Not okay.