Explanation: (1): subtract g times the fourth column from the first subtract h times the fourth column from the second (2): subtract e−g times the third column from the first subtract f−h times the third column from the second (3): subtract the fourth row from the second and third (4): subtract the third row from the first and second (5): expand on the third and fourth columns
..
cool
i think this will hold in general for $7 \times 7$, $3k+1 \times 3k+1$ matrices too
do you know what argument can we make for a general matrix?
will the above procedure 1-5 be independent of the size of the matrix?
This is done in three steps:
$H,K\unlhd G$ since each subgroup of an abelian group is normal.
Let $g\in H\cap K$. Since $g\in H$, we have $\phi(g)=g$, but $g\in K$, so $0=\phi(g)=g$. Thus $H\cap H=\{0\}$.
We have $$H+K=\{ h+k\in G\mid h\in H, k\in K\}\subseteq G.$$ Let $g\in G$. Consider $f=\p...
@Shaun Even those of us with lots of experience (e.g., robjohn and I) are repeatedly getting downvoted for no apparent reason. It's hopeless to try to figure it out. However, I would wish that someone explained to that OP that his $\phi(h)=h+K$ in his "attempt" is meaningless.
Reading a textbook right now, and the author states: “Let $f: S \rightarrow T$ be a function. We say that $f$ is invertible if there is a function $g: T \rightarrow S$ in the opposite direction having the property that $f \circ g(a) = a$ for all $a \in S$ and $g \circ f(c) = c$ for all $c \in $T$. Why is it "for all $a \in S$ and not all $a \in T$?
If $f\colon X\to Y$, then it really helps to write $f(x)=y$ and to pick elements' names accordingly. I don't deduct points for this, but when students chose $y\in X$, I made very mean comments.
With my first book (the algebra text), I was responsible for not only the typesetting but also the printing. I was very upset when I discovered, too late, that some of the fonts were not on the disk from which the computer and the printers printed. I was very upset.
By book #3, I expected small errors, even with my careful proofreading. The people who typeset my linear algebra text based on my LaTeX files redid it all and totally ruined it. By the second edition, I got back control.
The publisher hires editors to check basic things (and deal with stylistic language issues).
@wol If that's indicative of the level of care of the author, this is not going to be a pretty ride.
Generally, authors want you to inform them of mistakes or ambiguous things in their textbooks. They keep errata sheets — now, hopefully, updated regularly and posted on a webpage.
In time, there should be a corrected printing and another edition ... unless the book is not appreciated. There is no shortage of algebra texts and one always wonders what this one does that is so special and good.
Heya, I am self-studying Statistics, and running into several pockets of terminology that just seem to balloon into a never ending supply of extremely similar terminology, each term with its own Wikipedia page.
For example, there is Mean Square, Root Mean Square, Mean Square Error, Mean Square Deviation, Root Mean Square Deviation, Mean Absolute Deviation, Absolute Deviation, Square Deviation, Least Absolute Deviation, Minimum Least Mean Square Error, Median Absolute Deviation, Mean Percentage Error, Mean Absolute Percentage Error, Symmetric Mean Absolute Percentage Error, etc. The more pages I click on, the more associated terms I find.
Is this the sort of thing where people study each concept and gain specific intuitions for each, or is it good enough to just know that these are on Wikipedia to look up? Keeping 30 nearly identical terms straight sound incredibly tedious.
If @clarinetist ever shows up again, he's trained in statistics and has taught it and would definitely have some ideas. But, as I said, I haven't seen him in ages.
I suspect Youtube (not Wikipedia though) is partially displacing math textbooks as an explanatory tool, although textbooks may still be better for problems.
Sorry, nothing relevant to the question, some appear in a natural (as in physics, engineering) context, but often appear because they are easier to compute.
The statistician I knew in my grad school days at Berkeley (and I knew him because of bridge!) wrote a really good probability text, but not a stat text.
Well, that wouldn't surprise me. Do you write it for a 100-level course or do you write it with the geometric interpretations coming from all the linear algebra? But do most stat majors even know basic linear algebra? Ha.
When I first taught linear algebra my first quarter at UGA, I encountered linear regression as a projection problem (I had only seen it as a calculus exercise and didn't stop to think). When I asked a stat prof friend why they use least squares rather than other things, that friend didn't really have a good answer. I was fraught. "It makes the calculus easier." Wrong answer.
when i was an undergrad a good number of stat majors had a 'fear of math' problem similar to the problem that a good number of ed-oriented math majors had. it was weird, given the choice of major.
Yup. But you don't need much math to do undergraduate stat. The math ed issue is more weighty, since they're going to go wreak havoc on generations of kids.
I did encounter linear regression/least squares first in Linear Algebra. Most of the Statistics videos explaining it just say "here's what it basically is and here's how you compute it in Excel."
I actually found that when I created my algebra course with the math ed folks in mind (trying to tie in a bit more to the things in the high school curriculum rather than keeping it so abstract), a number of them had a much more positive attitude. I mean ... why do we factor? Why is an integral domain important? Ah.
On the other hand, leslie, I hope the stat major can explain why the Z-test tells you that things are more/less statistically significant. I certainly don't understand this.
my wife got an MS in stat. not as part of applying to a stat program but as an adjunct to another degree. the curriculum was "when you write papers, you're gonna wanna put numbers with decimal points in them, and this is how you do that."
i will cling to my premodern fear of numbers and my gut. you can't take those away from me. at least it's honest. i don't lie with Z-tests.
however, one sees stats applications when one gets to products of such volume that far out tails become relevant. not mentioning any of my nda restricted customers
hello, i wanted to clarify that it's true that if a set is countable, it forms a bijection with the natural numbers. this is y definition, right? so for proving a bijection exists between set A and the natural numbers, it's enough to show that set A is countable?
what is 'by definition' or not varies by your source, but yes, a common definition is that A is countable if there is a bijection between A and the natural numbers
and the most common way of working with that definition to prove that a set A is countable, is to construct a bijection between it and the natural numbers
some of the prose in your question is a little confusing to read. using that definition, you show that a set A is countable by constructing a function from A to N (or from N to A) and proving that the function you have constructed has the properties that make it a bijection
you need to define a function, any way of defining a function will do, as long as you can verify that whatever you are describing is a function. one of the most common ways to define a function is to give an explicit formula for it.
for example if we have a set {1,45,9} & {2,398,1000}, we cannot show these are both bijections with natural numbers without an explicit function? id say both of these can form a bijection but i could nto construct such a function to take 1->2 and 2->398, etc.
This is done in three steps:
$H,K\unlhd G$ since each subgroup of an abelian group is normal.
Let $g\in H\cap K$. Since $g\in H$, we have $\phi(g)=g$, but $g\in K$, so $0=\phi(g)=g$. Thus $H\cap H=\{0\}$.
We have $$H+K=\{ h+k\in G\mid h\in H, k\in K\}\subseteq G.$$ Let $g\in G$. Consider $f=\p...
