Let W be a subspace of $ \ \mathbb{R}^{n} \ $ .Show that $ \left \langle v,w \right \rangle =\left\langle (Proj)_{W}\ v ,w \right \rangle $

Question

Let W be a subspace of $ \ \mathbb{R}^{n} \ $ .Show that $ \left \langle v,w \right \rangle =\left\langle (Proj)_{W}\ v ,w \right \rangle $ , where $ Proj_{W} \ v$ means projection of v on the subspace W. $$ $$ Let $ \ \{u_{1},u_{2},...............,u_{n}\} $ be an orthogonal basis of W, then $ \ Proj(v,W)=\frac{\left\langle v,u_{1}\right \rangle}{\left\langle u_{1},u_{1}\right\rangle}u_{1}+................+\frac{\left\langle v,u_{n}\right\rangle}{\left\langle u_{n}\right\rangle}u_{n} , \ \ \ v \in \mathbb{R}^{n} \ and \ w \in W $ . But then how to approach. please help

Answer 1

Note that the basis of $W$ must contain less than $n$ vectors, otherwise $W=\mathbb{R}^{n}$. Let's call the number of elements of this base $k$. If $k=n$ then $Proj_Wv=v$ and the relationship is verified. If $k<n$, we can write an orthonormal basis of $\mathbb{R}^{n}$ starting from the basis of $W$: ${u_1,u_2,...,u_k,u_{k+1},...,u_n}$. You can use for example Gram-Schmidt process.Now you can write both $v$ and $w$ in terms of this basis. The coefficients of $w$ corresponding to $u_{k+1},...,u_n$ will be 0. Therefore, in the scalar product $\langle v,w\rangle$ the coefficients of $v$ corresponding to $u_j$ for $j>k$ will be multiplied by $0$.