Link to this document's Jupyter Notebook

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 on the day before class. Students must come to class the next day prepared to discuss the material covered in this assignment.


Readings for this topic (Recommended in bold)

Pre-Class Assignment: Vector Spaces

Goals for today's pre-class assignment

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  1. Basis Vectors
  2. Vector Spaces
  3. Lots of Things Can Be Vector Spaces
  4. Assignment Wrap-up

1. Basis Vectors

It is a really good review of concepts such as: Linear combinatins, span, and basis vectors.

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.

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 multiplication.)

Addition Axioms

  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. $1u = u$

3. Lots of Things Can Be Vector Spaces

Consider the following two matrices $A\in R^{3x3}$ and $B\in R^{3x3}$, which consist of real numbers:

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.

QUESTION (assignment specific): Determine whether $A$ is a linear combination of $B$, $C$, and $D$?

$$ A= \left[ \begin{matrix} 7 & 6 \\ -5 & -3 \end{matrix} \right], B= \left[ \begin{matrix} 3 & 0 \\ 1 & 1 \end{matrix} \right], C= \left[ \begin{matrix} 0 & 1 \\ 3 & 4 \end{matrix} \right], D= \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix} \right] $$

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!

Direct Link to Google Form

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: Is matrix $A$ is a linear combination of $B$, $C$, and $D$ from above?

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QUESTION: Summarize what you did in this assignment.

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

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

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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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Congratulations, we're done!

Course Resources:


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.