I found this cool quote that for now still blows my mind and seems cool although I am sure at a later point I will have a better grasp on things.
"A vector space that does not have a finite basis is called infinite dimensional. This is not an exotic property: many of the most important vector spaces, particularly spaces where the "vectors" are functions are infinite dimensional"
The parts that blew my mind were infinite dimensional not being an exotic property and the concept of functions as vectors. I am sure it will make more sense later.
Wednesday, March 26, 2014
Fields and Field extensions
huh \(\LaTeX\) what?
So I am reading through the section on fields and field extensions but I have to admit it is blowing my mind and I have having a hard time following the signicance of what is going on.
\( x>3 \) One example they give is the extension from \(\mathbb{R}\)(real numbers, I don't have latex setup to do fancy letters here). The polynomial P(x) = x^2 +1 has no real root. The root is i which isn't contained in R. We adjoin i to R and we get a new field C(complex) of the form a+bi with a,b contained in R.
One of the parts that got me(although writing this post is helping me think it through) was the concept of adjoining. I originally assumed that it meant you added that item into the set comprising the field but that doesn't seem to be the case. Adjoining does the funky thing the you see normally with complex numbers. If you adjoin item x, then the field had members of the form a+bx.
Another example is the field Q(rationals) which you can adjoin (2)^1/2 or the square root of 2.
The members of the new field are now of the form a + b*(2)^(1/2).
As a side note that looks really ugly I need to figure how to do nice latex and mathematical symbols on here if I want this to keep dumping funky math on this blog.
\( x>3 \) One example they give is the extension from \(\mathbb{R}\)(real numbers, I don't have latex setup to do fancy letters here). The polynomial P(x) = x^2 +1 has no real root. The root is i which isn't contained in R. We adjoin i to R and we get a new field C(complex) of the form a+bi with a,b contained in R.
One of the parts that got me(although writing this post is helping me think it through) was the concept of adjoining. I originally assumed that it meant you added that item into the set comprising the field but that doesn't seem to be the case. Adjoining does the funky thing the you see normally with complex numbers. If you adjoin item x, then the field had members of the form a+bx.
Another example is the field Q(rationals) which you can adjoin (2)^1/2 or the square root of 2.
The members of the new field are now of the form a + b*(2)^(1/2).
As a side note that looks really ugly I need to figure how to do nice latex and mathematical symbols on here if I want this to keep dumping funky math on this blog.
Wednesday, March 19, 2014
Chord
So I was just thrown for a loop by the phrase
"the derivative of a function f at a point x is the limit of the
gradients of a sequence of chords of the graph off"
It makes much more sense once I hit wikipedia http://en.wikipedia.org/wiki/Chord_(geometry)
The chord is just the line between two points and the circle and in this case the derivitive at a point is gradient or slope of the line as the chord shortens till that line is ifinetly small and we are essential dealing with a point.
We also see the sequence of chords since this is the derivative of a function rather than a single point.
Makes sense, the vocab though threw me off for a bit.
"the derivative of a function f at a point x is the limit of the
gradients of a sequence of chords of the graph off"
It makes much more sense once I hit wikipedia http://en.wikipedia.org/wiki/Chord_(geometry)
The chord is just the line between two points and the circle and in this case the derivitive at a point is gradient or slope of the line as the chord shortens till that line is ifinetly small and we are essential dealing with a point.
We also see the sequence of chords since this is the derivative of a function rather than a single point.
Makes sense, the vocab though threw me off for a bit.
Safari books online and Math
I have been interested in Safari books online for a number of years. I was first exposed during a summer internship in college.
I was working remote doing some web development and I came accross a need to use regular expressions. My manager at the time recommeneded "Mastering Regular Expressions" by Friedl and wow that is a good book. I was living with my parents and their local library had a subscription to safari online so that with my library card I was able read this book. After going back to school I spent a number of years as a poor college student and couldn't justify the cost of safari online.
Fortunately with a job change a few months ago I decided that regular daily investment in technical learning was a critical item in order for me to succeed in my software career.
I signed up and have been using safari online for several months now and find it very useful. In particular I have recently found "The princeton companion to mathematics" is available on safari.
At my last job at National Instruments there was a strong culture of books group and I was able to attend part of a book group based on the abstract algebra book found here.
http://abstract.ups.edu/download.html
I thoroughly enjoyed it and I hope to deepen and broaden my mathematical knowledge.
It is kinda funny as a kid I HATED math, it was repetitive and boring. I spent alot of effort(more on this another time) to take calculus in high school solely because it was required for Physics BC and I loved science.
Funny thing happened though. I loved calculus it was the first math course in which I felt that I was learning things that broadened my mind and helped me understand the world. It wasn't repetitive but based on simple concepts that could be extrapolated to further ideas.
Anyway I want to deeply understand modern mathematics especially so I can help make sure that my children see the beauty of math rather than rote repetition.
I will try and keep posting things here as I try and work my way through this thick and somewhat intimidating volume.
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