# Math Help - Basis

1. ## Basis

Let $B_1 = (v_1, v_2, v_3)$ be an ordered basis of a vector space V and let $B_2 = (w_1, w_2, w_3)$, where;

$w_1 = v_2 + v_3, w_2 = v_1 + v_3, w_3 = v_1 + v_2$

Verify that B2 is also a basis of V and find the change of basis matrices from B1 to B2 and from B2 to B1: Express the vector $av_1 + bv_2 + cv_3$
as a linear combination of $w_1, w_2 and w_3$

I dont really understand basis at all but my attempted answers so far are...

B2 is a basis of V because $w_1, w_2 and w_3$ are all linearly independent of V and have same dimension... maybe...

$av_1 + bv_2 + cv_3$ can surely just be expressed as any multiple of ${x}w_1 + {y}w_2 + {z}w_3$. For example if x=y=z=1 then a=b=c=2. Am i getting this bit totally wrong? It seems to simple an answer...

Any help with any of it would be could especially the change of basis bit.

2. The matrix $\frac{1}{2}\left[ {\begin{array}{rrr}{ - 1} & 1 & 1 \\ 1 & { - 1} & 1 \\ 1 & 1 & { - 1} \\ \end{array} } \right]$ will transform a vector in the ‘v-basis’ into a vector into the ‘w-basis’.

That matrix is the inverse of the matrix $\left[ {\begin{array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ \end{array} } \right]$.