18 Pre-Class Assignment: Inner Product

Goals for today’s pre-class assignment

  1. Inner Products

  2. Inner Product on Functions

  3. Assignment wrap-up

%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import sympy as sym
sym.init_printing()
---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
<ipython-input-1-f3e5c23067e5> in <module>
----> 1 get_ipython().run_line_magic('matplotlib', 'inline')
      2 import numpy as np
      3 import matplotlib.pyplot as plt
      4 import sympy as sym
      5 sym.init_printing()

~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/interactiveshell.py in run_line_magic(self, magic_name, line, _stack_depth)
   2342                 kwargs['local_ns'] = self.get_local_scope(stack_depth)
   2343             with self.builtin_trap:
-> 2344                 result = fn(*args, **kwargs)
   2345             return result
   2346 

~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/decorator.py in fun(*args, **kw)
    230             if not kwsyntax:
    231                 args, kw = fix(args, kw, sig)
--> 232             return caller(func, *(extras + args), **kw)
    233     fun.__name__ = func.__name__
    234     fun.__doc__ = func.__doc__

~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/magic.py in <lambda>(f, *a, **k)
    185     # but it's overkill for just that one bit of state.
    186     def magic_deco(arg):
--> 187         call = lambda f, *a, **k: f(*a, **k)
    188 
    189         if callable(arg):

~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/magics/pylab.py in matplotlib(self, line)
     97             print("Available matplotlib backends: %s" % backends_list)
     98         else:
---> 99             gui, backend = self.shell.enable_matplotlib(args.gui.lower() if isinstance(args.gui, str) else args.gui)
    100             self._show_matplotlib_backend(args.gui, backend)
    101 

~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/interactiveshell.py in enable_matplotlib(self, gui)
   3511         """
   3512         from IPython.core import pylabtools as pt
-> 3513         gui, backend = pt.find_gui_and_backend(gui, self.pylab_gui_select)
   3514 
   3515         if gui != 'inline':

~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/pylabtools.py in find_gui_and_backend(gui, gui_select)
    278     """
    279 
--> 280     import matplotlib
    281 
    282     if gui and gui != 'auto':

ModuleNotFoundError: No module named 'matplotlib'

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: $\(\langle (a_1,a_2),(b_1,b_2)\rangle = 2a_1b_1 + 3a_2b_2.\)$

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)

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

Assignment-Specific QUESTION: There is no Assignment specific question for this notebook. You can just say “none”.

Put your answer to the above question here

QUESTION: Summarize what you did in this assignment.

Put your answer to the above question here

QUESTION: What questions do you have, if any, about any of the topics discussed in this assignment after working through the jupyter notebook?

Put your answer to the above question here

QUESTION: How well do you feel this assignment helped you to achieve a better understanding of the above mentioned topic(s)?

Put your answer to the above question here

QUESTION: What was the most challenging part of this assignment for you?

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QUESTION: What was the least challenging part of this assignment for you?

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QUESTION: What kind of additional questions or support, if any, do you feel you need to have a better understanding of the content in this assignment?

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QUESTION: Do you have any further questions or comments about this material, or anything else that’s going on in class?

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QUESTION: Approximately how long did this pre-class assignment take?

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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.