You've probably noticed by now that the flavor of things happening in class and on the homework has changed since the first quiz. That's the nature of a Linear Algebra course - we start with simple ideas and things rapidly get more complex and abstract. To help you cope with all that, I present the

There are three main areas of preparation you want to work on:

Beyond that, here is Janeba's general guideline:

Note 1) Organizing the material in your head should usually
consume paper and ink. |

This is the part that you are probably most familiar with: learning
how to work through routine calculations. In this class, for instance,
we have learned how to solve a system of equations by *Gaussian elimination
*and *back substitution*, or by their close cousin, *Gauss-Jordan
reduction*. We have learned how to convert to and from matrix form.
We have just hinted at the process for computing the inverse of a matrix,
if one exists. We have learned how to add and multiply matrices. We have
learned a bunch of other things, too...

Of course you want to make sure you know the processes or algorithms
we have learned. It is probably helpful to make a list of them, though
sometimes you will have to invent your own name for a process, since we
haven't always given one. **The most important advice here** is that
the student must recognize that learning the algorithms is just a part
of what one must do to succeed in linear algebra.

In some ways, you could think of this "learning area" as being similar to learning the vocabulary in a language course. First of all, you have to spend some time doing the memorization. Also, as in language, there is more to just memorizing the words (or definitions): you need to know how things fit together.

For example, take the concept of *row echelon form*. You need to
know the definition, of course:

- The first nonzero entry of each row is 1.
- If row
*k*does not consist entirely of zeros, the number of leading zero entries in row*k*+1 is greater than the number of leading zeros in row*k*. - If there are rows whose entries are all zero, they are below the rows having nonzero entries.

DEFINITION: A matrix is said to be in row echelon form
if
from |

...but you need to have *digested* this a lot, also. Knowing just
the verbatim definition above wouldn't help much with a question like "*Give
examples of three distinct forms that the row-echelon form of a *3x3*
matrix could take".*

In addition to the formal definition of something, one should know:

- any alternate definitions that are easier to understand (like the ones we worked out in class)
- examples (for instance, of matrices in row-echelon form)
- "anti-examples" (in this case, examples of things that
**aren't**in row-echelon form - uses (what are things one can use row-echelon form for?)
- properties (things that are in row-echelon form are like such-and-such...)
- methods of proof, where applicable
- relevant theorems
- connections to other objects/ideas/concepts

Most importantly, you need to know how all this fits together. First
of all, that's what mathematics is all about - how things fit together
- which mathematicians call the *structure* of something. Secondly,
lots of psychological research has shown that it's far easier to remember
something that has some cohesive organization than something that is made
up of lots of unconnected pieces. Here's an image to give you the idea:

(Most of the ideas in this section are due to David Lay of the University
of Maryland. I learned them from a talk of his at the Joint AMS/MAA Meeting
in Baltimore this past January. In particular, the flower-metaphor and
building what he calls a *concept image* out of these various parts
are his ideas. Professor Lay requires his students to write out all these
components for each major concept in the course (though perhaps not in
flower form)).

So... why doesn't Prof. Janeba just give you a list of all this stuff?
First of all, *I* can't organize *your* thoughts. Secondly, if
someone wants to learn how to build a house, looking at someone else's
house can be a good idea, but practice will be much more helpful. So get
out your hammers, and start practicing!

As you know, this is not Willamette's main *learning-how-to-write-mathematical-proofs*
course, or at least not the first one that students take. In fact, though,
all advanced math courses include this topic. Just as in calculus one might
learn particular methods for computing certain integrals, in linear algebra
one learns *particular methods* for writing proofs pertinent to linear
algebra.

For example, we have learned a method for proving that one matrix is the inverse of another, i.e. for doing problems of the form:

Prove that ** (AB)^{-1 }= B^{-1}A^{-1}**
(assuming

Prove that ** (A^{T})^{-1} = (A^{-1})^{T}**
(assuming

Another example: we have already learned that proofs about matrices can come in two forms:

- proofs involving calculations on the entries in a matrix:

calculations like`a`_{ij}(b_{ij}+c_{ij})=`a`_{ij}b_{ij}+a_{ij}c_{ij}

- proofs that use the various properties of matrix algebra to avoid all
those subscripts,

for example: "`(A`^{-1})^{T}[...] and so`A`^{T}= (AA^{-1})^{T}= I^{T}= Iis the inverse of`(A`^{-1})^{T}."`A`^{T}

In ** all **proofs, one must have the definitions (see Learning
definitions and theorems) readily at hand. Beyond that, it will help
to consciously learn the particular proof techniques that we see in class
and in the readings. Making a list, in your own words, isn't a bad idea.

In all this, remember *Janeba's general guideline*.

*Last modified February 4, 1998.*

Prof. Janeba's
home page | *Please mail comments or questions to:* mjanebawillamette.edu

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