Question: if {v,w} is a base of a vector space V, then {v+w,v-w} is also a base of the vector space V. Prove?
need to know the steps to solve or any help would be nice. thanks
Because {v,w} is a basis they are linearly independent.
So suppose $\displaystyle \left( {\exists \alpha \wedge \beta } \right)\left[ {\alpha (v + w) + \beta (v - w) = 0} \right]$.
But that means that $\displaystyle \left( {\alpha + \beta } \right)v + \left( {\alpha - \beta } \right)w = 0$.
By independence we have $\displaystyle \left( {\alpha + \beta } \right) = 0\,\& \,\left( {\alpha - \beta } \right) = 0\quad \Rightarrow \quad \alpha = 0\,\& \,b = 0$
What does that prove?