11 Pre-Class Assignment: Vector Spaces
Contents
11 Pre-Class Assignment: Vector Spaces#
Readings for this topic (Recommended in bold)#
Goals for today’s pre-class assignment#
1. Basis Vectors#
Below is a really good review of concepts such as: Linear combinations, span, and basis vectors.
from IPython.display import YouTubeVideo
YouTubeVideo("k7RM-ot2NWY",width=640,height=360, cc_load_policy=True)
✅ QUESTION: What is the technical definition of a basis?
Put your answer to the above question here
✅ QUESTION: Write three basis vectors that span \(\mathbb{R}^3\).
Put your answer to the above question here
From the above video two terms we want you to really understand Span and Linear Independent. Understanding these two will be really important when you think about basis. Make sure you watch the video and try to answer the following questions as best you can using your own words.
✅ QUESTION: Describe what it means for vectors to Span a space?
Put your answer to the above question here
✅ QUESTION: What is the span of two vectors that point in the same direction?
Put your answer to the above question here
✅ QUESTION: Can the following vectors span \(\mathbb{R}^3\)? Why?
\((1,-2,3),\quad (-2,4,-6),\quad (0,6,4)\)
Put your answer to the above question here
✅ QUESTION: Describe what it means for vectors to be Linearly Independent?
Put your answer to the above question here
If you have vectors that span a space AND are Linearly Independent then these vectors form a Basis for that space.
Turns out you can create a matrix by using basis vectors as columns. This matrix can be used to change points from one basis representation to another.
✅ QUESTION: For each of the following subsets of \(\mathbb{R}^3\), determine if they (i) are linearly independent, (ii) span \(\mathbb{R}^3\), (iii) are bases of \(\mathbb{R}^3\):
(a) \(U=\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}2\\3\\0\end{bmatrix}\right\}\)
(b) \(V=\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}2\\3\\0\end{bmatrix},\begin{bmatrix}4\\5\\6\end{bmatrix}\right\}\)
(c)\(W=\left\{ \begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}2\\3\\0\end{bmatrix},\begin{bmatrix}4\\5\\6\end{bmatrix},\begin{bmatrix}7\\8\\6\end{bmatrix}\right\}\)
Put your answer to the above question here
2. Vector Spaces#
So far we have talked about vectors of real numbers (\(\mathbb{R}^n\)). However, there are other types of vectors as well. A vector space is a formal definition. If you can define a concept as a vector space then you can use the tools of linear algebra to work with those concepts.
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#
The sum \(u + v\) exists and is an element of \(V\). (\(V\) is closed under addition.)
\(cu\) is an element of \(V\). (\(V\) is closed under multiplication.)
Addition Axioms#
\(u + v = v + u\) (commutative property)
\(u + (v + w) = (u + v) + w\) (associative property)
There exists an element of \(V\), called a zero vector, denoted \(0\), such that \(u+0 = u\)
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#
\(c(u+v) = cu + cv\)
\((c + d)u = cu + du\)
\(c(du) = (cd)u\)
\(1u = u\)
3. Lots of Things Can Be Vector Spaces#
from IPython.display import YouTubeVideo
YouTubeVideo("YmGWj9RrNMI",width=640,height=360, cc_load_policy=True)
Consider the following two matrices \(A\in \mathbb{R}^{3x3}\) and \(B\in \mathbb{R}^{3x3}\), which consist of real numbers:
%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import sympy as sym
sym.init_printing()
a11,a12,a13,a21,a22,a23,a31,a32,a33 = sym.symbols('a_{11},a_{12}, a_{13},a_{21},a_{22},a_{23},a_{31},a_{32},a_{33}', negative=False)
A = sym.Matrix([[a11,a12,a13],[a21,a22,a23],[a31,a32,a33]])
A
b11,b12,b13,b21,b22,b23,b31,b32,b33 = sym.symbols('b_{11},b_{12}, b_{13},b_{21},b_{22},b_{23},b_{31},b_{32},b_{33}', negative=False)
B = sym.Matrix([[b11,b12,b13],[b21,b22,b23],[b31,b32,b33]])
B
✅ QUESTION: What properties do we need to show all \(3\times 3\) matrices of real numbers form a vector space.
Put your answer here
✅ DO THIS: Demonstrate these properties using sympy as was done in the video.
#Put your answer here.
✅ QUESTION: Determine whether \(A\) is a linear combination of \(B\), \(C\), and \(D\)?
#Put your answer to the above question here
✅ QUESTION: Write a basis for all \(2\times 3\) matrices and give the dimension of the space.
Put your answer to the above question here.
Written by Dr. Dirk Colbry, Michigan State University
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.