# 11 In-Class Assignment: Vector Spaces¶

## Agenda for today’s class (80 minutes)¶

1. (20 minutes) Review Pre-Class Assignment

2. (20 minutes) Introduction to subspaces

3. (20 minutes) Basis Vectors

4. (20 minutes) Vector Spaces

## 3. Basis Vectors¶

Consider the following example. We claim that the following set of vectors form a baiss for $$R^3$$:

$B = \{(2,1, 3), (-1,6, 0), (3, 4, -10) \}$

If these vectors form a basis they must be linearly independent and Span the entire space of $$R^3$$

%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import sympy as sym
sym.init_printing(use_unicode=True)

---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
<ipython-input-1-e3122a161773> in <module>
----> 1 get_ipython().run_line_magic('matplotlib', 'inline')
2 import matplotlib.pylab as plt
3 import numpy as np
4 import sympy as sym
5 sym.init_printing(use_unicode=True)

~/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'


DO THIS: Create a $$3 \times 3$$ numpy matrix $$A$$ where the columns of $$A$$ form are the basis vectors.

#Put your answer to the above question here

from answercheck import checkanswer



DO THIS: Using python, calculate the determinant of matrix $$A$$.

# Put your answer to the above question here.


DO THIS: Using python, calculate the inverse of $$A$$.

# Put your answer to the above question here.


DO THIS: Using python, calculate the rank of $$A$$.

# Put your answer to the above question here.


DO THIS: Using python, calculate the reduced row echelon form of $$A$$.

# Put your answer to the above question here.


DO THIS: Using the above $$A$$ and the vector $$b=(1,3,2)$$. What is the solution to $$Ax=b$$?

#Put your answer to the above question here.

from answercheck import checkanswer



Turns out a matrix where column vectors are formed from basis vectors a lot of interesting properties and the following statements are equivalent.

• The column vectors of $$A$$ form a basis for $$R^n$$

• $$|A| \ne 0$$

• $$A$$ is invertible.

• $$A$$ is row equivalent to $$I_n$$ (i.e. it’s reduced row echelon form is $$I_n$$)

• The system of equations $$Ax = b$$ has a unique solution.

• $$rank(A) = n$$

Not all matrices follow the above statements but the ones that do are used throughout linear algebra so it is important that we know these properties.

## 4. Vector Spaces¶

A Vector Space is a set $$V$$ of elements called vectors, having operations of addition and scalar multiplication defined on it that satisfy the following conditions ($$u$$, $$v$$, and $$w$$ are arbitrary elements of $$V$$, and c and d are scalars.)

### Closure Axioms¶

1. The sum $$u + v$$ exists and is an element of $$V$$. ($$V$$ is closed under addition.)

2. $$cu$$ is an element of $$V$$. ($$V$$ is closed under scalar multiplication.)

1. $$u + v = v + u$$ (commutative property)

2. $$u + (v + w) = (u + v) + w$$ (associative property)

3. There exists an element of $$V$$, called a zero vector, denoted $$0$$, such that $$u+0 = u$$

4. For every element $$u$$ of $$V$$, there exists an element called a negative of $$u$$, denoted $$-u$$, such that $$u + (-u) = 0$$.

### Scalar Multiplication Axioms¶

1. $$c(u+v) = cu + cv$$

2. $$(c + d)u = cu + du$$

3. $$c(du) = (cd)u$$

4. $$1u = u$$

### Definition of a basis of a vector space¶

A finite set of vectors $${v_1,\dots, v_n}$$ is called a basis of a vector space $$V$$ if the set spans $$V$$ and is linearly independent. i.e. each vector in $$V$$ can be expressed uniquely as a linear combination of the vectors in a basis.

## Vector spaces¶

DO THIS: Let $$U$$ be the set of all circles in $$R^2$$ having center at the origin. Interpret the origin as being in this set, i.e., it is a circle center at the origin with radius zero. Assume $$C_1$$ and $$C_2$$ are elements of $$U$$. Let $$C_1 + C_2$$ be the circle centered at the origin, whose radius is the sum of the radii of $$C_1$$ and $$C_2$$. Let $$kC_1$$ be the circle center at the origin, whose radius is $$|k|$$ times that of $$C_1$$. Determine which vector space axioms hold and which do not.

### Spans:¶

DO THIS: Let $$v$$, $$v_1$$, and $$v_2$$ be vectors in a vector space $$V$$. Let $$v$$ be a linear combination of $$v_1$$ and $$v_2$$. If $$c_1$$ and $$c_2$$ are nonzero real numbers, show that $$v$$ is also a linear combination of $$c_1v_1$$ and $$c_2v_2$$.

DO THIS: Let $$v_1$$ and $$v_2$$ span a vector space $$V$$. Let $$v_3$$ be any other vector in $$V$$. Show that $$v_1$$, $$v_2$$, and $$v_3$$ also span $$V$$.

### Linear Independent:¶

Consider the following matrix, which is in the reduced row echelon form.

$\begin{split} \left[ \begin{matrix} 1 & 0 & 0 & 7 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 3 \end{matrix} \right] \end{split}$

DO THIS: Show that the row vectors form a linearly independent set:

DO THIS: A computer program accepts a number of vectors in $$R^3$$ as input and checks to see if the vectors are linearly independent and outputs a True/False statment. Discuss in your groups, which is more likely to happen due to round-off error–that the computer states that a given set of linearly independent vectors is linearly dependent, or vice versa? Put your groups thoughts below.