/sci/ - Science & Math

https://en.wikipedia.org/wiki/Gershgorin_circle_theorem

>>9648816
mods

>>9648816
I don't like how confused I am by chapter 1 section 1.2 of Hoffman and Kunze. Linear combinations just feel like silly equation smashing to solve for scalar solutions of the linear combinations.

Linear combinations were presented briefly with almost no motivation. Additional resources would be stellar.

>>9649009
Supplement it with

https://www.amazon.com/Linear-Algebra-Step-Kuldeep-Singh/dp/0199654441

He contains detailed solutions to every exercise in the book for free on a website. He really breaks Linear Algebra down into a "step-by-step" approach.

>>9649062
>https://www.amazon.com/Linear-Algebra-Step-Kuldeep-Singh/dp/0199654441
Thanks. I've been bouncing around between several. Valenza's text was a tad too abstract, but I do really admire the organization of the content (starting with underlying structures of algebra [groups, subgroups, monoids, semigroups, magmas, fields, rings, rngs, etc and delves into kernels, homomorphisms, etc all by page 20). I think I'll also open up LADR later today, as I'm beginning to agree with the pedagogy within (omitting determinants until the last chapter) despite that initial oddity being why I decided against it.

Thank god for libgen.

>>9649096
Yeah, no problem. LADR is also abstract. Starts with vector spaces almost immediately. It's a great book, but assumes it's your second exposure to LA.

The book I recommended is for the first semester course in LA and is in the same realm as Strang's Linear Algebra book. It's more of an applied approach vs theoretical. Within its realm (first course in LA textbook) it's better than Strang.

>>9649114
Is an abstract approach pedagogically unsound for a first exposure? I suppose, to answer my own question, that it would come down to one's ability to understand abstract concepts (aka maths maturity).

>be me, physics/cs joint major
>get to quantum
>all the physics majors are baffled by matrix representations of operators
lol

>>9649131
Give LADR a shot. If you don't understand it, move to the book recommended here: >>9649096 then complete that book and move back to LADR.

>>9649142
The book mentioned here comment is referring to: >https://www.amazon.com/Linear-Algebra-Step-Kuldeep-Singh/dp/0199654441 -- I think that book would provide good intuition for Axler.

how do i prove pic related? i think i have the first part right, but i dont know where to go now.
i know that somehow the RHS sum implies (along with m>n) that there are non-zero b_i such that the LHS sum is zero, but ive no idea in what way

>>9649009
nigga i believe you are retarded. what else do you think one does to solve linear equations? did you miss high school? we got taught this when we were at least 14: there's three methods to solve a linear equation: substitution, graphing, and elimination. Elimination is the only one that is consistent, and it is precisely equation mashing. You don't need motivation for linear combinations lmao. Here's one if you want:

John has x bananas and 2y pineapples which cost him 2$, and Alice has 3x bananas and y pineapples which cost him 3$. How many bananas and pineapples did john and alice buy???

What's everyone's favorite vector space which has no basis?

>>9649169
Use basis

>>9649249
R^3

>>9649249
the one im using to encode cryptographic information

>>9648816
This is the best way to learn linear algebra:
graphicallinearalgebra.net

The inverse of a matrix is the reciprocal of the determinant times the adjugate of the cofactor matrix

At the end of the day, its the only field (finite dimensional vector spaces) that isnt contriversial with any math autist.

>>9649169

>>9649282
yeah it is

>>9649232
Everything in math needed motivation to be discovered. I enjoy rediscovering the magic of it, but understand the perspective of those that don’t.

I just didn’t like how cut and dry it was presented, I didn’t say I was struggling with the algebra of it. I imagine later on the authors will paint a more vivid picture of its uses through its relationships and uses in further topics.

>>9649142
I’m likely going to supplement Valenza and H&K with LADR. Thanks for the recommendations, if all else fails I’ll use the text you recommended.

>>9648825
That's pretty interesting actually. Thanks

>>9649249
R over Q

i like it cause it lets me do econometrics

>>9649311
Thanks.
I think i'm misunderstanding a few things. Doesn't [math] \sum c_j v_i=0\Longleftrightarrow c_j=0 [/math] mean the vectors are linearly [math] \textit{in} [/math]dependent? Also I don't get how it follows from m>n.
>>9649009
>>9649431
they come up a bit more when vector spaces are introduced, which i think is the next chapter, but for now theyre just expressions. i guess the motivation, if there is any, is that they are simple and have nice properties, plus pretty much everything later on involves them in some way.

Linear algebra is quite boring until you introduce a topology.

I like it, but I think it is completely stupid to do operations by hand. Fuck that.

>>9649148
>>9649142
Got it, thanks!

>>9649613
Can you elaborate. Interested to hear this as I don't yet know topology.

What is the best book to continue learning linear algebra? I've taken "Basic Linear Alegbra" at my uni so I have exposure to matrices, determinants, vector spaces, linear transformations, and eigens but that's really about it. I will be taking "Standard Linear Algebra" next semester but want to prepare a bit beforehand.

My linear algebra professor taught us some very interesting stuff about Google's page rank algorithm. It involves matrices apparently.

My favorite part was when I finished the final exam and walked out of that classroom for the last time.

Why is the determinant so powerful? Everything I've learned so far in Lin.Alg can be simplified immensely with a determinant

>>9650179
3brown1blue linear algebra series. Axlers LADR. LADR assumes you mastered your first course in LA and understand more abstract areas of mathematics. But it starts “basic” in the sense it defines what scalars, vectors, vector spaces area in the first part of the book as if you’ve never seen them before.

>>9650179
As a follow up there is another good Linear algebra book that’s technically a first book in linear algebra but goes into some good theoretical detail at the same time. I’ll try to look it up and (you) you it later

>>9648816
MODS

>>9649632
Kek

What are your favorite linear algebra text?

>>9650918
LADR

>>9650383
because it's determined to make it

>>9650918
Valenza's

>>9650918
Friedberg Insel Spence

>>9650383
Because its a function that assigns a real number to every matrix, and its much easier to work with real numbers than matrices

>>9651889
>Because its a function that assigns a real number to every matrix
Never post again until you finish at least one semester of linear algebra

>>9651889
does that make [math] f\colon\mathscr{M}_{m,n}(\mathbb{R})\longrightarrow\mathbb{R}\,; f\colon(a)_{ij}\longmapsto a_{11} [/math] powerful?

>>9651889
>A real number to every matrix
I regret asking this question.

On the upside though I just learned alg.geom and the other algebra (forget what it's called) and it's relationship with eigens

>>9651709
I’ll check the text out

Pls help

>>9653060
what is your question

>>9653073
vector AB' is rotated to AB''
we have
X' = AB''cos(m+n)
X'= (Xi+Yj)cos(m+n)
= Xi(cos(m)cos(n) - sin(m)sin(n))+Yj(cos(m)cos(n)-sin(m)sin(n))

I tried writing all individual vector components but final vector is not coming aligned with X'

>>9648825
pls help me
>>9653060

>>9653092
Bumping for your question.

>>9648996
>>9650688
what did he mean be this

>>9653811
Protecting his virginity

>>9650331
Thought it was the sum of all probabilities that led up to a specific site.

>>9654502
probabilities of what?
i always thought it was along the lines of

[math] \text{page rank rating of site } i =\sum\limits_{k\neq i}\text{links from site } k \text{ to site } i\cdot\text{page rank rating of site }k [/math],
which i suppose you could then write with matrices as the other anon mentioned

bump

>>9649249
Depends what you mean by "has a basis". All vector spaces have bases, but not all can be constructed.

>>9650383
It's a super nice invariant that's compact and tells you a bunch.

For instance, the sign if the determinant tells you the orientation of a transformation. It also is a homomorphism into your field, so you know it works across multiplication. The scalar value of your determinant tells you by what factor it scales volumes.

>>9652838
it’s good, innit?

>>9650145
Various linear subgroups of nxn matrices have cool topological properties. GL(2,R) is topologically a 4-manifold, and SL(2,R) is a submanifold.

In particular, SL(2,R) are matrices in GL(2,R) with determinant 1. Determinant is a smooth function, and R is a 1-dim manifold, so SL(2,R) is a smooth manifold of codimension of a 1 dimensional submanifold, so dimension 3.

Which is kewl. SL(2,R) is a dimension 3 submanifold.

Idk what the other guy was referring to, probably an undergrad trying to sound smart.

>>9649289
Path integral formulation of QM takes integrals over function spaces with infinite bases, and measure theory treats R as a vector space over Q of uncountable cardinal, which basically creates the entire foundation for modern probability theory.

But go ahead, pretend no math exists outside of your intro physics classes.

>>9655892
>It also is a homomorphism into your field
No, it's a group homomorphism.

>>9655892
>Homomorphism
Explain. We looked over isomorphisms. To be fair we were taught the determinant is n-multilinear. Not sure if it has to do with that since we never proved it.
Yeah the sign of the determinant is a change of basis orientation so I would imagine the whole standard basis axis is flipped over. T

>>9656046
>Explain.
det(AB)=det(A)det(B)

>>9656046
he meant determinant is a homomorphism from GL(n) to non-zero real numbers
as a function of vectors, it is n-linear. thus it is a homomorphism if considered as a mapping from the n-th tensor power of R^n to real numbers, if you wanna be fancy

How would you prove that all the bases of an infinite dimensional vector space V have the same cardinality? Here's what I would do: if A and B are two bases, each a in A can be expressed as finite sum of elements of B, so the cardinality of the subset of B which contains the span of A = V is [math] \leq |A| \times \aleph_0 = |A| [/math]. This subset must exhaust B or else there would be a b in B which could be expressed as a linear combination of other elements in B, contradicting the fact that B is a basis. So [math] |B| \leq |A|
[/math] and by symmetry, |B| = |A|.

>>9656051
>non-zero real numbers
Determinant maps to any base field (or ring), not just reals

>>9648816
i like when u multiply two matrixes

>>9656191
I like it when you do the add together

>>9656191
.

>>9649613
Not really. IP spaces are plenty cool for statistical learning and signal processing applications.

This guy is cool:
https://www.youtube.com/watch?v=usvDP8x2MS8&list=PL774A268635BA8AAD

Feel like I am watching a UFC fighter teach math

>>9649062
>amazon link
http://download.library1.org/get/3DB0C8D83D35D77D10024BD510E95F5F/Kuldeep%20Singh-Linear%20Algebra_%20Step%20by%20Step-Oxford%20University%20Press%20%282013%29.pdf

>>9660843
This