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Pre-Class Assignment: Inner Product

Goals for today's pre-class assignment

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  1. Inner Products
  2. Inner Product on Functions
  3. Assignment wrap-up

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 $c$:

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

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?

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3. Assignment wrap-up

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Written by Dr. Dirk Colbry, Michigan State University Creative Commons License
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