In order to successfully complete this assignment, you must do the required reading, watch the provided videos, and complete all instructions. The embedded survey form must be entirely filled out and submitted on or before 11:59pm on the day before class. Students must come to class the next day prepared to discuss the material covered in this assignment.
Pre-Class Assignment: Vector Spaces¶
Goals for today's pre-class assignment¶
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1. Basis Vectors¶
It is a really good review of concepts such as: Linear combinatins, 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 $R^3$.
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 $R^3$? 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.
2. Vector Spaces¶
Vector spaces are an abstract concept used in math. So far we have talked about vectors of real numbers ($R^n$). 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.
✅ **DO THIS:** Skim Chapter 2 II pg 77-86 of the Heffron text and Chapter VS pg 257-269 of the Beezer text to get a feel for the ideas around Vector spaces and use these as a reference if you get confused.
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¶
- $c(u+v) = cu + cv$
- $(c + d)u = cu + du$
- $c(du) = (cd)u$
- $1u = u$
from IPython.display import YouTubeVideo
YouTubeVideo("YmGWj9RrNMI",width=640,height=360, cc_load_policy=True)
Consider the following two matrices $A\in R^{3x3}$ and $B\in R^{3x3}$, 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 3:** 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 $D$?
$$ A= \left[ \begin{matrix} 7 & 6 \\ -5 & -3 \end{matrix} \right], B= \left[ \begin{matrix} 3 & 0 \\ 1 & 1 \end{matrix} \right], C= \left[ \begin{matrix} 0 & 1 \\ 3 & 4 \end{matrix} \right], D= \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix} \right] $$#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.
5. Assignment wrap-up¶
Please fill out the form that appears when you run the code below. You must completely fill this out in order to receive credit for the assignment!
If you have trouble with the embedded form, please make sure you log on with your MSU google account at googleapps.msu.edu and then click on the direct link above.
✅ **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?
Put your answer to the above question here
✅ **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?
Put your answer to the above question here
✅ **QUESTION:** Do you have any further questions or comments about this material, or anything else that's going on in class?
Put your answer to the above question here
✅ **QUESTION:** Approximately how long did this pre-class assignment take?
Put your answer to the above question here
from IPython.display import HTML
HTML(
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Congratulations, we're done!¶
To get credits for this assignment, you must fill out and submit the above survey form on or before the assignment due date.
Course Resources:¶
Written by Dirk Colbry, Michigan State University
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.