# 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 (

__–____\right ) \\ \quad \quad = \frac{1}{4} \left (____+____+__ + – __–__

__– – \right ) \\ \quad \quad = \frac{1}{4} \left (____+__ – __– \right ) \\ \quad \quad = \frac{1}{4} \left (__

__+__ + __+__ \right ) \end{align}

- Since $V$ is an inner product space over the real numbers, we have that $
__= \overline{__and thus:} = $

(2)

\begin{align} \quad \quad = \frac{1}{4} \left ( 4

__\right ) =____\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 \|^2 + \| v \|^2 + __+__ \end{align}

(4)

\begin{align} \quad \quad – \| u – v \|^2 = –

__= –____–____– – = – \| u \|^2 – \| v \|^2 +____+__ \end{align} (5)

\begin{align} \quad \quad \| u + iv \|^2 i =

__i =____i +____i +__i + i = \| u \|^2i + \| v \|^2i + __–__ \end{align}

(6)

\begin{align} \quad \quad -\| u – iv \|^2 i = –

__i = –____i –____i – i -i = – \| u \|^2i – \| v \|^2i +____–__ \end{align} - If we sum these quantities up, we get that $\| u + v \|^2 – \| u – v \|^2 + \| u + iv \|^2i – \| u – iv \|^2 i = 4
__$__. Therefore:

(7)

\begin{align} \quad

__= \frac{ \| u + v \|^2 – \| u – v \|^2 + \| u + iv \|^2i – \| u – iv \|^2 i}{4} \quad \blacksquare \end{align}__### Related post:

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