Let $v_1=(1,1,0,3), v_2=(1,2,1,1)$ and $v_3=(0,2,1,0)$ be vectors of $\Bbb{R^4}$ and $V=\text{span}(v_1,v_2,v_3).$ Determine if $v=(5,7,8,0)$ belongs in $V$.
So I would need to reduce the following matrix $$\begin{bmatrix}v_1 && v_2&&v_3 \mid v \end{bmatrix}.$$
However this matrix seems to be inconsistent so in conclusion $v \notin V$?
Best Answer
Yes, that will work. Or you can compute the determinant of the matrix$$\begin{bmatrix}1 & 1 & 0 & 5 \\ 1 & 2 & 2 & 7 \\ 0 & 1 & 1 & 8 \\ 3 & 1 & 0 & 0\end{bmatrix}.$$Since it is not $0$ (actually, it is $-13$), the four vectors are linearly independent and therefore $v$ is not a linear combination of the other three.