Given ${u, v, w}$ is a basis for $\mathbb{R}^3$, how can I show that $\{u + v + w, v + w, w\}$ is also a basis?
I solved a similar problem in $\mathbb{R}^2$ (or at least think I did :p).
Given $\{u, v\}$ is a basis for $\mathbb{R}^2$, show that $\{u + v, au\}$ is also a basis.
I used the definition of a span to say
$\implies cu + dv =\langle x, y\rangle $ (for $c, d$ in $\mathbb{R}$)
Let $c = d + a^2$ (for $a$ in $\mathbb{R}$)
$\implies (d + a^2)u + dv = \langle x, y\rangle $
$\implies d(u + v) + a(au) = \langle x, y\rangle $
$\implies \mathrm{span}(u + v, au) = \mathbb{R}^2$
From there I also showed that this set was linearly independent (by starting with putting $c$ and $d$ equal to zero) and concluded that, as it had both properties in $\mathbb{R}^2$, it must be a basis.
So I've been wracking my brains trying to find ways to manipulate coefficients to achieve the new basis for $\mathbb{R}^3$, but I can't come up with anything.
I'd like to know if the method I employed for the $\mathbb{R}^2$ question is acceptable? Is this the only way to do it? Is there another method that I should be using?
P.S I apologise for my imperfect formatting. Still learning. Somehow I kept collapsing all the spaces between symbols.