Formulas for The Inner Product

# Formulas for The Inner Product

We will now look at two alternate formulas for computing the inner product between two vectors in a vector space. Theorem 1 gives us a formula for real inner product spaces, while Theorem 2 gives us an analogous formula for complex inner product spaces.

 Theorem 1: Let $V$ be an inner product space over the real numbers. Then for all $u, v \in V$ we have that $= \frac{ \| u + v \|^2 - \| u - v \|^2}{4}$.
• Proof: If we expand $\frac{\| u + v \|^2 - \| u - v \|^2}{4}$ we have that:
(1)
\begin{align} \quad \quad \frac{\| u + v \|^2 - \| u - v \|^2}{4} = \frac{1}{4} \left ( \| u + v \|^2 - \| u - v \|^2 \right ) = \frac{1}{4} \left ( <u + v, u + v> - <u - v, u - v> \right ) \\ \quad \quad = \frac{1}{4} \left ( <u, u> + <u, v> + <v, u> + <v, v> - <u, u> - <u, -v> - <-v, u> - <-v, -v> \right ) \\ \quad \quad = \frac{1}{4} \left ( <u, v> + <v, u> - <u, -v> - <-v, u> \right ) \\ \quad \quad = \frac{1}{4} \left ( <u, v> + <v, u> + <u, v> + <v, u> \right ) \end{align}
• Since $V$ is an inner product space over the real numbers, we have that $<u, v> = \overline{<v, u>} = <v, u>$ and thus:
(2)
\begin{align} \quad \quad = \frac{1}{4} \left ( 4 <u, v> \right ) = <u, v> \quad \blacksquare \end{align}
 Theorem 2: Let $V$ be an inner product space over the complex numbers. Then for all $u, v \in V$ we have that $= \frac{ \| u + v \|^2 - \| u - v \|^2 + \| u + iv \|^2i - \| u - iv \|^2 i}{4}$.
• Proof: Let's look at each term in the numerator separately.
(3)
\begin{align} \quad \quad \| u + v \|^2 = <u + v, u + v> = <u, u> + <u, v> + <v, u> + <v, v> = \| u \|^2 + \| v \|^2 + <u , v> + <v, u> \end{align}
(4)
\begin{align} \quad \quad - \| u - v \|^2 = -<u - v, u - v> = -<u, u> - <u, -v> -<-v, u> - <-v, -v> = - \| u \|^2 - \| v \|^2 + <u, v> + <v, u> \end{align}
(5)
\begin{align} \quad \quad \| u + iv \|^2 i = <u + iv, u + iv>i = <u, u>i + <u, iv>i + <iv, u>i + <iv, iv>i = \| u \|^2i + \| v \|^2i + <u, v> - <v, u> \end{align}
(6)
\begin{align} \quad \quad -\| u - iv \|^2 i = -<u - iv, u - iv>i = -<u, u>i -<u, -iv>i - <-iv, u>i -<-iv, -iv>i = - \| u \|^2i - \| v \|^2i +<u, v> - <v, u> \end{align}
• If we sum these quantities up, we get that $\| u + v \|^2 - \| u - v \|^2 + \| u + iv \|^2i - \| u - iv \|^2 i = 4<u, v>$. Therefore:
(7)
\begin{align} \quad <u, v> = \frac{ \| u + v \|^2 - \| u - v \|^2 + \| u + iv \|^2i - \| u - iv \|^2 i}{4} \quad \blacksquare \end{align}