i like typing out things
whereby i have absolutely no understanding whatsoever
it is a way of knowing
a way of saying
i do not know what this means
and there is a very good chance that
i will never know
a matrix is non-singular
(meaning that it has an invese matrix)
if and only if
the identity matrix
can be obtained
using only elementary row
operations.
this is a principle of
gauss-jordan elimination
i am struggling to know why this could be important
yet i find a strange delight in being able
to articulate this sentence as a poem
and a testimony to what must be
a first principle of epistemology
"to know that i do not know"
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10 comments:
jh—
So, you're reading linear algebra now?
i am struggling to know why this could be important
Matrices are representations of linear transformations over finite-dimensional vector spaces. An informative example of a finite-dimensional vector space would be the Euclidean plane over the reals.
We can imagine the points of the Euclidean plane as having a few simple algebraic properties:
1. c (x,y) = (c x, c y) for all real c, x, y
2. (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)
The first property is called "scalar multiplication," and the second "addition." A linear transformation is a function T that interacts nicely with this vector space structure, i.e.
1. T (c v) = c T(v) for all real c and all v = (x,y)
2. T(v + w) = T(v) + T(w).
For example, consider the matrix
-1 0
0 1
This represents a reflection around the y-axis, i.e.,
T(x,y) = (-x,y)
Now, there are basically three kinds of linear transformations on the Euclidean plane:
1. T could map everything to the point (0,0).
2. T could map everything to a line, e.g., the matrix
0 0
0 1
represents the transformation T(x,y) = (0,y), i.e., every point is projected horizontally onto the y axis.
3. T maps onto the entire plane. If so, T is 1-1 and its corresponding matrix is (technically, matrices are) non-singular.
You can think of the importance this way. Often, in practical mathematical modeling, we're given T(x,y) for some linear transformation T, and we need to find (x,y). This is simply a technical version of the ontological problem: "How did I get here?"
For non-singular systems, there is alway a unique "right" answer to that question.
The most remarkable thing to me about this is the venue.
ah yes
i'd like to be able to say
OK now i get it
no disrespect intended
your explanation
is more mysterious
than the inital post
i am only amazed that
this sort of language
happens
jh
gauss-jordan elimination
i think i learned that once
it all sounds so familiar
just not quite fully accessible to me
i do understand the usefulness though
i am currently reviewing a paper
on which i am a co-author
that uses linear algebra
the goal is to use GPS measurements
of the slow, tectonic motion
of specific points on the ground
to infer the slip-rates of active faults
i helped to collect the GPS data
and can help the geophysicists
who are the primary authors
to ensure that their fault models
fit with what is known of the geology
but i will leave the matrix inversion
to their capable hands
i must admit
that i never expected
to read about gauss-jordan elimination
on this blog
jh—
More words do not necessarily result in greater clarity, especially in technical discourse. I think this is as true of theology as mathematics.
Sally—
Gauss-Jordan elimination is the technique I learned as the "augmented matrix method"
The idea is pretty simple. Given an n x n matrix A, adjoin the n x n identity matrix I, forming AI, an n x 2n matrix.
Then, you perform elementary row operations on AI (i.e., interchanging rows, or subtracting a multiple of one row from another). The goal is transform the matrix to IB. Once you've done this, B is A inverse.
Proving that this is correct -- that takes a bit more work.
thanks for the pragmatix sally
i thought well maybe it helps
with flight computers or
satellite TV or something
i still am at the point of extreme hilarity trying to understand this
it seems so funny to me
but of course
epistemology requires a very wide berth
somehow i feel there's an advent message of hope
in this
maybe if i read it one more time
and try not to be amused
jh
i learned a new term this past summer
from students
from the texting generation:
mego--"my eyes gloss over"
that was my initial reaction
to stu's careful explanations
but once i was willing to expend
a little bit of effort i found
i was indeed able to wrap my brain
around it again
(just don't ask me to explain it myself)
the phrase gauss-jordan is so familiar to me
i think that must have been the term
that my linear algebra teacher used
when i was in college
i think hilarity
is a much more healthy repsonse
than mego
prepare ye the way of the lord
fill in those potholes
invert those matrices
make way for the coming savior
i have to say
that the christmas concert
at our church last sunday
felt really anticlimatic to me
like now that we've sung the whole story
christmas must be over
before it has even started
i kept thinking to myself
why are we singing this?
this is not the right time
i've never had that sense before
maybe i'm beginning
to understand advent
never before have i conceived that linear algebra might resonate with themes of gaudete REJOICE
i'm thinking of a new O antiphon
O identity matrix o interstices before the convex parabolic three dimensional idiomatic tranverse duplicating free radical spectrum vortex orbital transmigration of souls onto celestial infinities
will there be linear algebra in heaven
i sure want to talk to that guass guy when i get there
i want to be on the space ship lord and i want those servants of god manning those identity matrices
my matrix
sounds like
something you might call your mother
if you were talking about her and you were a little drunk
let me introduce you to
my matrix
i really think i am carrying some ancient anger against Euclid
adn that englishman who started this rectangualr mapping of everything putting these euclidean grids down...grids grids grids everywhere straight lines where things really ought to be curved....after reading through the explanation once again stu i realize
i'm really quite angry with Euclid
i'm not at all sure that identity matrix has anything to do with my anger but Euclid is starting to piss me off...in another 1000 yrs i'll be steaming
brilliant conversation
3X = 9
X + 3 = 12
I'm not sure in what way that is true, or why, but I'm pretty sure it checks out.
Everything above this is really mysterious. But I am still going on my Cliff's Notes Algebra 1.
I'm like the red dot on a YouTube video trying desperately to stay ahead of the screaming red line behind me which represents my childrens' understanding of mathematics.
Oh, I meant 3X + 3 = 12.
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