Let V = Span{(1, 2, 3), (1, 0, 2)} and W = Span{(0, 2, 1), (1, 1, 1)}. Determine the relation between V and W , i.e. whether V ⊆ W , or W ⊆ V , or neither.
May I ask how should I approach this question? I have found the reduced row echelon form of the 2 spans by equating them to (x,y,z). Not sure whether I have taken the correct approach but I'm not sure how to draw a relation between them.
Any help is appreciated, thanks!
Here is a general method. To show $V\subseteq W$ you need to show that both $(1,2,3)$ and $(1,0,2)$ are in $W$, that is, both vector equations $$x_1(0,2,1)+x_2(1,1,1)=(1,2,3)\quad\hbox{and}\quad x_1(0,2,1)+x_2(1,1,1)=(1,0,2)$$ have solutions. For $W\subseteq V$, same but the other way round.
However since we are in $\mathbb R^3$, here is a short cut. Both $V$ and $W$ are planes through the origin. You can use the cross product to calculate the normal to each. If these normals have the same direction then the planes are the same. If not, then neither is a subset of the other.