Let $ w_1,w_2,\dots,w_n$ be an orthonormal basis of $W$. If $v = a_1\cdot w_1+a_2\cdot w_2+\dots+a_n\cdot w_n$, then $a_1 = ?\ a_2=?\ a_n = ?$

Question

Let $w_1, w_2,\dots, w_n$ be an orthonormal basis of $W$. If $v = a_1\cdot w_1+a_2\cdot w_2+\dots+a_n\cdot w_n$, then $a_1 = ?\ a_2=?\ a_n = ?$

How do I find the scalars $a_1, a_2$, and $a_n$ ?

Answer 1

Since $W^{-1} = W^T$: $Wa=v \iff W^TWa=W^Tv \iff a=W^Tv$