Anybody here good at graduate/4th year linear algebra? We're currently on norms and inner products and the shit is bonkers hard. Pic related is the book we're using. If I give you a couple of exercises (or personal questions, that I just don't know), could you give me some help?
>>7950699
I can show you how to stare at matrices all day and just see how fucking cool they look.
>>7950699
shoot at me as a nigger you are. MY BODY IS READY FOR YOUR QUESTIONS.
>>7950764
> basic linear algebra
Bro, this isn't row reduction to Jordan Normal form or finding eigenvalues/eigenvectors (well, we did do that, but it was a lot more complex than the more basic, junior level course).
>>7950737
My biggest question is 8.3 here (pic related). What I have so far for part a is that for any t on the interval [a,b] there exists functions, f(t) and g(t) in U and a constant, c, such that f(t) + c = g(t), so U is closed under addition. I did a similar thing for multiplication to prove that it's closed under multiplication (because U contains all continuous (over some interval, [a,b]), complex functions, so if c = 5 and f(t) = 2t (over [0,4], f(t) + c = 2t + 5 = g(t) which is in U. My biggest issue is parts B and C. I'm thinking for part B you show that ||f|| is a normed space (using the equation given) with the three criteria for being a normed space? Then C you do the same thing for proving <f,g> is an inner product using the formula given (and the 4 criteria for being an inner project)? I mean, I kind of have an idea of hat I'm doing here, but want to know if I'm on the right track.
i read that this was the best intro to linear algebra
this is a fucking grad school text? did I get memed on?
How's Axler's book?
>>7950872
A lot of graduate linear algebra classes use it. If you use chapter 1-3 or 4 it's a pretty good intro to upper level linear algebra (my school calls it Linear Algebra I and it's a junior level math class). My class is Linear Algebra II, which is a senior level class. That said, there is a graduate class that uses the book too. Honestly, the book kind of has everything. It starts off with vector spaces, stuff leading up to the rank-nullity theorem, and then into eigenvalues and eigenvectors (more complicated than just the mechanics of how to do it, this book focusing on invariant subspaces, direct sums, and the Jordan Decomposition theorem). My class will probably end at around chapter 10 or 12 (there are 23 chapters).
The book would not be good for an intro linear algebra (the freshman level class where you just work with basic, linear systems and culminates in the method of finding eigenvalues and eigenvectors).
I'm not familiar with Axler's book
>>7950863
yes you just check the definition of the concepts you want to prove.
>>7950872
serge land is best book
>>7950863
I believe C is a consequence of B. I know that linear transformations that are norm preserving are also inner product preserving and visa versa.
>>7950863
Son, (a) follows from the fact, that if f, g are cont. functions af+bg is a cont. function. Now since the addition here is just pointwise addition function h=0 is the 0 element of U.
(b) You need to show that ||f|| is finite on U and that ||*|| satisfies the definition of the norm. The positive homogenity of the norm and the fact that the norm separates points is trivial, the triangle inequality follows frorm Minkowski inequality.
(c) is straightforward.
>>7950863
rational and JNF are way harder than this. this is just "verify they fulfill conditions". you should have used this norm before, right? in analysis? that's just the corresponding inner product. come on, this is groundwork.
>>7950939
I had a junior level class that had some analysis. It was called advanced calculus and was some topology and some real analysis. It started off with basic set theory, moving to stuff like open, closed and disconnected sets, and culminating in proving the mean value theorem, intermediate value theorem, and fundamental theorem of calculus. We never really dealt with norms in the class. I could have taken either real analysis or complex analysis, but chose not to because I didn't like advanced calculus. I did most of my in major focus (we had to select a sequence of courses) on differential equations (PDEs and nonlinear diff eqs) and mathematical modeling (population dynamics and a course on chaos). I'm not saying this is super difficult, linear algebra has always been kind of an achilles' heel for me. I just wanted to make it clear that this isn't the freshman level linear algebra class that you can take without even doing single variable calculus.
>>7950917
Does showing that af + bg is continuous for continuous functions, f and g, show that U is a vector space over natural addition and multiplication or do I need to show that for any f in U, af is also in U?
>>7950980
>Does showing that af + bg is continuous for continuous functions, f and g, show that U is a vector space?
Yes
>>7951007
Ok, now finally, how would I go about showing that the norm is greater than or equal to 0? I know it does because the part inside the integral is the square of the absolute value of the function which seems intuitive enough, but how would I go about showing this?
>>7950980
>Does showing that af + bg is continuous for continuous functions, f and g, show that U is a vector space
Yes
>do I need to show that for any f in U, af is also in U?
Once you show that [math]af+bg \in \mathcal{U},[/math] you can set [math]b = 0,[/math] as [math]0 \in \mathbb{C},[/math] and thus [math]af \in \mathcal{U}.[/math] You don't need to make that statement separately.
>>7951086
Note that [math]f^2>0[/math] everywhere, so [math]\int f^2 > 0.[/math]
>>7950872
Axler's book is more geared towards an undergraduate learning how to do proof based mathematics. Harry Dym's book has more content and assumes a higher mathematical maturity level. That said, while I do recommend Axler's book, I have my own personal issues with it.
>>7950863
haha holy shit i did this in first year along with all the first year maths students
>>7950872
http://www.maa.org/press/maa-reviews/linear-algebra-in-action-0
>>7951007
Can you read a definition you fucking faggot?
>>7950863
This is completely trivial, just check the definitions.
