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Topic: Failing Linear Algebra:
Replies: 54   Last Post: Jan 10, 2007 12:47 PM

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Russell Blackadar

Posts: 586
Registered: 12/12/04
Re: Failing Linear Algebra:
Posted: Apr 27, 2004 8:24 PM
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On 27 Apr 2004 12:13:44 -0700, Anonymous wrote:

>rusin@vesuvius.math.niu.edu (Dave Rusin) wrote in message news:<c69sdb$oi7$1@news.math.niu.edu>...
>> In article <c98b1ba0.0404221444.4623535e@posting.google.com>,
>> Anonymous wrote:

>> >I'm currently a math major and am taking linear algebra, but I'm in
>> >serious danger of failing. I just don't get it! Is this newsgroup a
>> >place to come to ask questions and get information about learning
>> >math? Or is there somewhere more appropriate to go? I've always had
>> >trouble with vectors, and I think I fell apart sort of right at the
>> >beginning of linear algebra (although, I did manage to get a B- on the
>> >very first exam). I've got another exam next week. What can I do? I
>> >don't get all the terms, concepts, and jargon. Anyone know how to
>> >make learning linear algebra easier and more practical? Anyone got
>> >any practice problems?

>>
>> I would bet that the single practice problem you need to work on is,
>> "What is a vector space?"
>>
>> Our students "do well on the very first exam" because that's the part
>> of the course where we warm up with techniques for solving linear
>> systems of equations and such topics. But our LA course is also our
>> students' first course in which abstractions, axioms, and proofs play
>> a significant role. They often stumble because (among other problems)
>> they don't realize they need to _memorize_ definitions _precisely_.
>> So you can do a little self-assessment here to figure out whether
>> what you're missing is bits of topics or the core idea: can you,
>> right this minute, define what a vector space is?
>>
>> dave

>
>Dave, this may be my problem. I did decent on the first exam because,
>like you said, it was solving linear systems, echelon form, linear
>dependences--easier stuff like that. I've always been a little
>confused with the concepts, but I think I may finally be getting a
>grip on what a vector space is:
>
>It's a group of vectors that can be multiplied by any scalar and/or
>added together in any way,


Not just any way -- there are precise conditions that the sum must
satisfy. Look for the axiomatic definition in your book(s). It's not
at all difficult to memorize. Do it.

and whatever possible combinations that can
>result is the "vector space" for that group of vectors. This is how I
>understand it. For vectors in R^2, a plane is formed ("spanned"???)
>by the vector space. For vectors in R^3, a solid area is formed by
>the vector space. It gets difficult for me to move into dimension 4.


Geometric examples are often helpful, but use them to *liberate*
yourself, not tie yourself down. If you are faced with a vector in
R^4 on your exam, don't bother trying to visualize it. You can do
everything you need to do algebraically.

Personally, I keep a few geometric examples from R^2 in my head, and
don't bother with anything more complicated unless the problem happens
to be specifically about geometry, which won't be the case on your
exam this week. E.g. the matrix for a 90-degree rotation in R^2, in
the usual basis, is useful to remember -- or even better, you might
learn how to derive it quickly on the fly -- because if (say) you find
yourself confused about how basis vectors transform, you have a ready
example to check out and remind yourself. Other 2x2 matrices for
simple rotations and reflections (and perhaps more) are good to know,
but not for their own sakes, but rather because they help you think
clearly about the abstractions (and only if they truly do).

>While I understand that the same concepts hold, there's no more
>physical picture I can use to visualize what's happening. Is my
>understanding of "vector space" sufficient enough? Am I missing
>anything?


The algebra is the important thing, not the picture. And your current
understanding of the algebra is insufficient to keep you from getting
confused on the upcoming test. You need to learn the definitions
precisely.

>
>I know I am still struggling with the concepts of "span" and "basis".
>The weird thing is that I'm alright with the more advanced stuff;
>matrices, determinants, eigenvalues, eigenvectors. I'm a little hazy
>with diagonalization because it's the newest thing we've done. I know
>it's got something to do with the eigenvalues of a special type of
>matrix.
>
>I guess I could also really use some help with understanding how a
>mapping gets converted into a matrix, and then how to solve it.


I like the term "linear transformation" and I think you should use it
too; a "mapping" usually means something more general that may not
have the necessary restrictions. Definition time again! Do you know
what restrictions I'm talking about? f(a+b) = f(a)+f(b) and
f(ca) = cf(a) of course. That, by definition, is what makes the
mapping *linear*.

Anyhow, you ask about how a linear transformation is converted into a
matrix. The l.t. is the abstract thing; the matrix is one particular
representation of it (in terms of two bases). Change one or both of
the bases, and you get a different matrix for the same linear
transformation. It's somewhat analogous to the way you can write the
same abstract number (say, the cube of two) as 8 in decimal, or 10 in
octal, or 1000 in binary. But better than that, I think, because all
of the important features of the linear transformation have
counterparts in matrix theory; i.e. whatever matrix you end up using,
it will share some important features with the transformation, e.g. it
will have the same eigenvalues, same rank, etc.

(So, one can say that the set of nxm matrices is isomorphic to the set
of linear transformations from F^n to F^m. Indeed this is an
isomorphism in the strict sense I give below, because both these sets
are vector spaces in their own right! But don't worry about that if
you find it confusing.)

In answer to your question, you aren't *really* converting the
transformation into a matrix -- they are two different things -- but
for most practical purposes you can conveniently ignore that fine
distinction

OTOH if what you're asking is how (by what method) to do this
conversion, that actually is quite easy. Take basis vector #1, apply
the transformation to it componentwise to get the components of the
transformed vector, and write those components in a column. Do the
same with basis vector #2, writing it as a column to the right of the
one you already wrote. And so on, until you're done. There's your
matrix. Try working it out for the 90-degree rotation I mentioned
above, and then try your matrix out on some 2D column vectors to see
if they really do turn 90 degrees when you multiply by the matrix.

(I am assuming you multiply with matrix on the left and column vector
on the right, as is done in Schaum's outline; hopefully that is how
your prof does it too, otherwise swap positions and take the
transposes!)

>I
>understand matrix multiplication and can do it well. But the concepts
>of image, kernel, and isomorphism and how they relate to the
>mappings/matrices seem to be lost on me. The odd thing is that I
>fully understand the definitions of "kernel" and "image" as they were
>applied in algebraic structures, but I don't get how they apply to
>linear really. "Isomorphism" is a concept I never understood in
>algebraic structures or linear algebra.


If you get it for algebraic structures in general, you are ahead of
the game. In the context of linear algebra, an isomorphism is simply
a linear transformation (remember the definition?) that has an
inverse. Kernel and image have their usual meaning.

(Isomorphisms come up a lot in math; see above. In general they tell
you when it's OK to substitute one structure by another, without
losing any essential information.)

One nice thing about linear algebra is that once you have a matrix for
a linear transformation, you can answer questions e.g. about the
l.t.'s kernel and image very easily, simply by manipulating the
matrix, without having to wrestle with the abstraction. If the kernel
is {0}, i.e. the set of rows (or the set of columns) of the matrix is
linearly independent, and the matrix is square, then the matrix is
invertible and you have an isomorphism. Otherwise, one or more rows
or columns is a linear combination of the others; and that means there
are nonzero vectors which, when multiplied by your matrix, give zero.
In other words, those vectors are members of the kernel of your linear
transformation. See how it all fits together?

Btw you asked elsewhere what an eigenvalue is. The definition is
simplicity itself, learn it! If your question is really, what are
they good for, one answer is that they come up a lot in differential
equations. A part of math that linear algebra has a lot to say about.

>
>Thanks to the people who posted book suggestions, but I'm hesitant
>about buying any other books. I already bought the text, the Cliff's
>Notes guide to linear algebra, a 2003-version of the Schaum's outline,
>and I even have an old 1968 version of Schaum's that my grandmother
>used when she majored in math. Cliff's has been helpful, but too
>basic. Schaum's seems almost too advanced; it's great that they solve
>all the problems, but sometimes the explanations are lacking. I find
>that I do much better at math problems if I can first figure out how
>to solve a certain type of problem and then go back and try to
>understand the concepts behind it, rather than the other way around.
>Schaum's examples don't allow for this, because they assume you've
>already read (and understood) the concepts behind how to solve certain
>problems.


Yes, I think you have enough books. Personally, I like the Schaums
Outline (the standard one by Lipschutz) a lot, by the way. As one who
easily gets confused myself, I can say it was the book that made the
quickest sense to me, i.e. had the smallest eyes-glaze-over effect.
OTOH, I did have the benefit of having read and worked on some other,
more abstract books beforehand. Btw I found nothing objectionable
about Lipschutz from an abstract point of view; if he made it any
simpler he would mislead, but he treads that fine line very nicely
IMHO.

>
>Maybe, if it isn't too much to ask, would anyone here be willing to
>post some problems relating to mappings/kernel/image/isomorphims
>and/or eigenvalues/eigenvectors, and I can attempt to solve them with
>your help?


Learn the definitions first, vet them with some examples and
counterexamples as some others have recommended, and then focus on
learning some important techniques cold, and with luck you can ace the
test (and be a math major too, and even be happy about it someday).
Usenet will probably be too slow for your needs this week; who wants
to type in those matrices in ASCII anyway.


Date Subject Author
4/22/04
Read Failing Linear Algebra:
Guest
4/22/04
Read Re: Failing Linear Algebra:
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1/10/07
Read Re: Failing Linear Algebra:
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1/10/07
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1/10/07
Read Re: Failing Linear Algebra:
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4/23/04
Read Re: Failing Linear Algebra:
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4/23/04
Read Re: Failing Linear Algebra:
Brian Borchers
4/27/04
Read Re: Failing Linear Algebra:
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1/10/07
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maky m.
4/26/04
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4/28/04
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4/28/04
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4/29/04
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4/28/04
Read Re: Failing Linear Algebra:
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1/10/07
Read Re: Failing Linear Algebra:
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4/29/04
Read Re: Failing Linear Algebra:
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4/29/04
Read Re: Failing Linear Algebra:
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1/10/07
Read Re: Failing Linear Algebra:
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1/10/07
Read Re: Failing Linear Algebra:
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5/1/04
Read Re: Failing Linear Algebra:
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1/10/07
Read Re: Failing Linear Algebra:
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1/10/07
Read Re: Failing Linear Algebra:
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1/10/07
Read Re: Failing Linear Algebra:
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4/30/04
Read Re: Failing Linear Algebra:
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1/10/07
Read Re: Failing Linear Algebra:
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1/10/07
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4/27/04
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4/30/04
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