1.3 The Dot, oops, INNER Product

1

Using spike diagrams, go over the steps of taking the dot product of the vectors (1,2) and (2,4).

2

Define the dot product of two functions $f(x)$ and $g(x)$ that are defined for $0<x<1$.

3

In what sense is the dot product between the earlier two vectors similar to the inner product between the functions $x$ and $2x$ defined for $0<x<1$? In what fundamental sense is it different?

4

What do you need to do with dot products and inner products if your vectors respectively functions become complex? If you do so, what can you say about a dot/inner product of a vector/function with itself?

5

When are vectors/functions orthogonal to each other?

6

Give three vectors in three dimensional space that form an orthonormal set.

7

Give a set of two orthonormal functions defined on the interval $0<x<1$.