The dot product is not the only sensible way to measure lengths and angles. Any product obeying the same three rules works just as well, in any vector space, and everything proved from those rules carries over: lengths, angles, orthogonality and all that follows.

Distances on a map can be measured in miles or in hours of walking, which weighs uphill paths more heavily. Both are honest measures, and both obey the same rules of distance.

An inner product on a vector space $V$ gives every pair of vectors a number $\langle\mathbf{u},\mathbf{v}\rangle$ that is symmetric, linear in each slot, and positive: $\langle\mathbf{v},\mathbf{v}\rangle>0$ for every $\mathbf{v}\neq\mathbf{0}$. Lengths, distances, angles and orthogonality are then defined from it exactly as from the dot product. Every proof so far in this chapter used only these three rules, so Cauchy–Schwarz, Pythagoras and the independence of orthogonal vectors hold for every inner product.

• The dot product on $\mathbb{R}^n$, and weighted versions such as $\langle\mathbf{u},\mathbf{v}\rangle=2u_1v_1+3u_2v_2$ on .

Write $\mathbf{u}$ and $\mathbf{v}$ in a basis $B$ and expand the inner product by linearity in each slot. What is left is a double sum of the coordinates weighted by the numbers $\langle\mathbf{b}_i,\mathbf{b}_j\rangle$, which is a matrix product. That matrix , the Gram matrix, is symmetric because the inner product is; for the dot product and the standard basis it is .

Write both vectors in the basis.

Linearity in each slot expands the inner product term by term.

That double sum is the product of a row, a matrix and a column.

$\langle\mathbf{b}_i,\mathbf{b}_j\rangle=\langle\mathbf{b}_j,\mathbf{b}_i\rangle$, so is symmetric.

A formula is an inner product when it is symmetric, linear in each slot, and positive on every non-zero vector. The last is the one most often missed.

On $\mathbb{P}_2$, $\langle p,q\rangle=p(0)q(0)+p(1)q(1)+p(2)q(2)$ multiplies the values at , and and adds, like a dot product of the lists of values.

• An inner product is symmetric, linear in each slot and positive on non-zero vectors.
• Length, angle and orthogonality come from it as from the dot product: $\|\mathbf{v}\|=\sqrt{\langle\mathbf{v},\mathbf{v}\rangle}$.
• Examples: weighted dot products, sums of values of polynomials, .

Great work! You now know how to:

• check the three rules of an inner product;
• compute inner products on vectors, polynomials and matrices;
• use lengths and orthogonality relative to an inner product;
• write an inner product as its Gram matrix.