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Section 4.2 Definition and Properties of an Inner Product

We have already discussed the concept of a dot product Section 3.6 or inner product Section 3.7 in the simplest examples of vector spaces when we can think of the vectors as either arrows in space or as columns of real or complex numbers. In this section, we generalize the concept of inner product to ANY vector space.

An inner product \(\left\langle \vec{u}\vert \vec{v}\right\rangle\) is a generalization of the dot product with the following properties:

\begin{align} \left\langle \vec{u}\vert \vec{v}\right\rangle \amp = \left\langle \vec{v}\vert \vec{u}\right\rangle^*\tag{4.2.1}\\ \left\langle \vec{u}\vert \lambda\vec{v}+\mu\vec{w}\right\rangle \amp=\lambda\left\langle \vec{u}\vert \vec{v}\right\rangle +\mu\left\langle \vec{u}\vert \vec{w}\right\rangle\tag{4.2.2} \end{align}

Like the dot product, the inner product is always a rule that takes two vectors as inputs and the output is a scalar (often a complex number).

The existence of an inner product is NOT an essential feature of a vector space. A vector space can have many different inner products (or none).

In analogy with inner products, we call the square root of the inner product of a vector with itself \(\braket{v}{v}\) the norm or the length of the vector. Similarly, if the inner product of two vectors is zero \(\braket{v}{w}=0\) we say that the vectors are orthogonal or perpendicular even when these statements have no obvious geometric meaning.