Given B =\begin{pmatrix} 1 & -1 & 1\\ 1 & 1 & 3\end{pmatrix}determine the general solution of the homogeneous system Bx=0. This describes the intersection of two objects (e.g. a line or a plane). Determine and find the Cartesian equation of this object.
I have solved the homogeneous equation being x$=t(-2, -1, 1)$ but how do I determine what two objects this intersection represents along with their Cartesian equation?
Note that the first two columns are linearly independent, this means rank (dimension of the range space) is at least $2$. But range is $\subseteq \Bbb{R}^2$. Thus the range is entire $\Bbb{R}^2$. By the rank-nullity theorem, the nullity is $3-2=1$. This means the solution space for $B\mathbf{x}=\mathbf{0}$ is $1$ dimensional. Thus it must be a line passing through the origin.
Note: we didn't have to really solve the system to determine the geometry of the solution space.
From your answer it is clear that any solution is parallel to the vector $\begin{bmatrix}-2\\-1\\1\end{bmatrix}$. Hence it is line whose direction vector is this and passes through the origin.