Suppose that $\{u, v, w\}$ is a linearly independent set of vectors in $\mathbb{R}^{50}$. Show that Span$\{u, v, w\}$ = Span$\{u − v, u − 2v + w, v + w\}$.
I know that in order to do this, I need to prove that each span is a subset of the other. So far, I have: $A :=$ Span$\{u, v, w\}$ and $B := $Span$\{u − v, u − 2v + w, v + w\}$. Then, I showed that:
- $u-v = -1(u)-1(v)+0(w)$ so $b1 ∈ A$.
- $u-2v+w = 1(u)-2(v)+1(w)$ so $b2 ∈ A$.
- $v+w = 0(u)+1(v)+1(w)$ so $b3 ∈ A$.
Since any element $b$ in $B$ can be written as a linear combination of $b1, b2$, and $b3, B$ is a subset of $A$. I struggle with how to prove that $A$ is a subset of B, though. Guess and check is becoming very tedious.