Math 3323 Matrix Algebra
Investigation 2
Fall 2007
Note!: As of now (4pm, Sunday), the webMathematica page is not working properly.
(So it's not your browswer or your computer...)
I've put in a call to our computer systems person, but I don't know how soon she will get to it.
Feel free to try the webMathematica even before I update this page, just in case the problem gets
fixed in between the times I'm checking. But certainly, as soon as I hear things are fixed, I'll
post that news here.
Update: The webpage is working!!!
Open this Mathematica page in a new browser window, so you can see both these instructions and the Mathematica page at the same time.
Main goal
Your main goal is to find as many different "rules"
as possible to help someone determine if a set of vectors (in
3-dimensional space) is linearly independent or linearly dependent.
Your rules should not depend on the components of the vectors
involved.
Technical reminders
Remember that you can plug in linear combinations of vectors, for instance, 5a1-7a2. You can even enter vectors by specifying their components (for instance, {3,-2,4}), but remember that your rules should not depend on the components of the vectors involved.
Remember, also that you can test to see if a set of vectors is linearly dependent with a particular linear combination by entering that linear combination as one of the vectors, and seeing if it comes out to be the zero vector. There isn't a comparable way to prove a set of vectors is linearly independent, but you still may be able to convince yourself that a given set of vectors is linearly independent.
Experiments
Let's start with two vectors: Enter the vectors a1 and a2 into the boxes. Do you think the set is linearly independent? If not, state at least one linear combination among the vectors that reveals the linear dependency of the set. Next try the same thing with a1 and a4. Now try your own examples of two vectors. Can you give any descriptions of when a set of two vectors is linearly dependent? Can you give any descriptions of when a set of two vectors is linearly independent?
Now we'll go on to three vectors: Enter the vectors a1, a2, and a3 into the boxes. Do you think the set is linearly independent? If not, state at least one linear combination among the vectors that reveals the linear dependency of the set. Next try the same thing with a1, a2, and a9.
Repeat with the sets of vectors {a1,a4,a5}, and then {a1,a2,a9}, and then {a3,a4,a8}.
Try your own examples of three vectors. Can you give any descriptions of when a set of three vectors is linearly dependent? Can you give any descriptions of when a set of three vectors is linearly independent?
Finally, let's try four vectors: The instructions are the same as above with two and three vectors. You can start with {a1,a2,a3,a4} and {a1,a2,a3,a6}. Note that you can't display all 4 vectors and the linear combination of them you are trying to show independence, so this is harder. But do the best you can.
Report
Write a summary of your observations and conjectures. Describe all the
experiments that you did. List all the conjectures you made (including
any that later turned out to be false), how you came up with them, how
you tested them, and which ones you still think are true. Keep in mind the main goal of this investigation. Please use
clear, complete sentences, and organize your report.