Quote:

Suppose that $\displaystyle v_1, v_2,......,v_n$ are orthogonal non-zero vectors in Euclidean n-space, and that a vector v is expressed as

$\displaystyle v=\lambda_1 v_1 + \lambda_2 v_2+.......+\lambda_n v_n,$

Show that the scalars $\displaystyle \lambda_1, \lambda_2,......., \lambda_n$ are given by

$\displaystyle \lambda_i=\frac{v.v_i}{||v_i||^2}, i=1,2,.....,n$

What are $\displaystyle \lambda_i$ if the vectors $\displaystyle v_1,v_2,.........,v_n$ are orthonormal?

Well this is what i'm stuck on. The first step of my method was to multiply both sides by$\displaystyle v_i$. However, the question seems to hinge on what v has to be equal to. I can't see what this is. My initial thought was v=0, but this does not make sense since the vectors can have different magnitudes.