17 Pre-Class Assignment: Inner Product#

Goals for today’s pre-class assignment#

  1. Inner Products

  2. Inner Product on Functions

%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import sympy as sym
sym.init_printing()

1. Inner Products#

Definition: An inner product on a vector space \(V\) (Remember that \(R^n\) is just one class of vector spaces) is a function that associates a number, denoted as \(\langle u,v \rangle\), with each pair of vectors \(u\) and \(v\) of \(V\). This function satisfies the following conditions for vectors \(u, v, w\) and scalar \(c\):

  • \(\langle u,v \rangle = \langle v,u \rangle\) (symmetry axiom)

  • \(\langle u+v,w \rangle = \langle u,w \rangle + \langle v,w \rangle\) (additive axiom)

  • \(\langle cu,v \rangle = c\langle u,v \rangle\) (homogeneity axiom)

  • \(\langle u,u \rangle \ge 0 \text{ and } \langle u,u \rangle = 0 \text{ if and only if } u = 0\) (positive definite axiom)

The dot product of \(R^n\) is an inner product. Note that we can define new inner products for \(R^n\).

Norm of a vector#

Definition: Let \(V\) be an inner product space. The norm of a vector \(v\) is denoted by \(\| v \|\) and is defined by:

\[\| v \| = \sqrt{\langle v,v \rangle}.\]

Angle between two vectors#

Definition: Let \(V\) be a real inner product space. The angle \(\theta\) between two nonzero vectors \(u\) and \(v\) in \(V\) is given by:

\[cos(\theta) = \frac{\langle u,v \rangle}{\| u \| \| v \|}.\]

Orthogonal vectors#

Definition: Let \(V\) be an inner product space. Two vectors \(u\) and \(v\) in \(V\) are orthogonal if their inner product is zero:

\[\langle u,v \rangle = 0.\]

Distance#

Definition: Let \(V\) be an inner product space. The distance between two vectors (points) \(u\) and \(v\) in \(V\) is denoted by \(d(u,v)\) and is defined by:

\[d(u,v) = \| u-v \| = \sqrt{\langle u-v, u-v \rangle}\]

Example:#

Let \(R^2\) have an inner product defined by: $\(\langle (a_1,a_2),(b_1,b_2)\rangle = 2a_1b_1 + 3a_2b_2.\)$

✅ QUESTION 1: What is the norm of (1,-2) in this space?

Put your answer to the above question here.

✅ QUESTION 2: What is the distance between (1,-2) and (3,2) in this space?

Put your answer to the above question here.

✅ QUESTION 3: What is the angle between (1,-2) and (3,2) in this space?

Put your answer to the above question here.

✅ QUESTION 4: Determine if (1,-2) and (3,2) are orthogonal in this space?

Put your answer to the above question here.

2. Inner Product on Functions#

from IPython.display import YouTubeVideo
YouTubeVideo("8ZyeHtgMBjk",width=640,height=360, cc_load_policy=True)
# https://www.youtube.com/watch?v=8ZyeHtgMBjk

Example#

Consider the following functions

\[f(x)=3x-1\]
\[g(x)=5x+3\]
\[\text{with inner product defined by }\langle f,g\rangle=\int_0^1{f(x)g(x)dx}.\]

✅ QUESTION 5: What is the norm of \(f(x)\) in this space?

Put your answer to the above question here. (Hint: you can use sympy.integrate to compute the integral)

✅ QUESTION 6: What is the norm of g(x) in this space?

Put your answer to the above question here.

✅ QUESTION 7: What is the inner product of \(f(x)\) and \(g(x)\) in this space?

Put your answer to the above question here.

Written by Dr. Dirk Colbry, Michigan State University Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

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Congratulations, we’re done!#