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Pre-Class Assignment: Inner Product¶

Goals for today's pre-class assignment¶

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Inner Products
Inner Product on Functions
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 $c$:

$\langle u,v \rangle = \langle v,u \rangle$ (symmetry axiom)
$\langle u+v,w \rangle = \langle u,w \rangle + \langle v,w \rangle$ (additive axiom)
$\langle cu,v \rangle = c\langle v,u \rangle$ (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: $$\langle (x_1,x_2),(y_1,y_2)\rangle = 2x_1y_1 + 3x_2y_2.$$

✅ **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?

Put your answer to the above question here.

3. Assignment wrap-up¶

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✅ **Assignment-Specific QUESTION:** There is no Assignment specific question for this notebook. You can just say "none".