How do I describe geometrically all linear combinations of (1,2,3) and (3,6,9)? (these are columns)
When I figure this out I just get 3 vectors in the x, y and z coordinates, each of which doubles in length. These combine to form a single line.
However, other combinations like (1,0,0) and (0,2,3) "form a plane". How can a single line in 3d space form a plane? Why does this linear combination form a plane when the previous one doesn't?
Since $$(3,6,9)=3(1,2,3)$$
the linear combinations decribe a line in $\mathbb{R^3}$ through the origin.