Orthogonal Basis Expansion

Theorem

If $S=\{{\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_n\}$ is an orthogonal basis of an inner product space $V$, and if $\mathbf{u}$ is any vector in $\mathbf{v}$, then

$$\mathbf{u}=\frac{⟨\mathbf{u},{\mathbf{v}}_1⟩}{\Vert {\mathbf{v}}_1{\Vert }^2}{\mathbf{v}}_1+\frac{⟨\mathbf{u},{\mathbf{v}}_2⟩}{\Vert {\mathbf{v}}_2{\Vert }^2}{\mathbf{v}}_2+\cdots +\frac{⟨\mathbf{u},{\mathbf{v}}_n⟩}{\Vert {\mathbf{v}}_n{\Vert }^2}{\mathbf{v}}_n$$

If $S=\{{\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_n\}$ is an orthonormal basis of an inner product space $V$, and if $\mathbf{u}$ is any vector in $\mathbf{v}$, then

$$\mathbf{u}=⟨\mathbf{u},{\mathbf{v}}_1⟩{\mathbf{v}}_1+⟨\mathbf{u},{\mathbf{v}}_2⟩{\mathbf{v}}_2+\cdots +⟨\mathbf{u},{\mathbf{v}}_n⟩{\mathbf{v}}_n$$