18 Pre-Class Assignment: Inner Product¶

Goals for today’s pre-class assignment¶

Inner Products
Inner Product on Functions
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”.