VECTOR SPACES - LINEAR ALGEBRA

We introduce vector spaces in linear algebra.
#LinearAlgebra #Vectors #AbstractAlgebra
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In this video we talk about Vector Spaces and ask ourselves if some sets are vector spaces. We also talk about the polynomial vector space.
Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any questions, leave them below. I try to answer as many questions as possible. If something isn't quite clear or needs more explanation, I can easily make additional videos to satisfy your need for knowledge and understanding.

[@nirupamayyagari6571]

Ive been told so much that linear algebra focuses more on visualizing the actual spaces rather than functions. Ive watched videos and read material and somewhat understood it, but your drawings and this video and the functions you chose made that make so much more sense.

[@beedlejeedle]

dude you have no idea how much the visual representation helped; why my prof never explained this visually is beyond me

[@jared9190]

Helpful video. I was having a little trouble with this concept but your explanation and visuals were just what I needed.

[@ruskodudesko9679]

agreed man - you may be a visual learner, I know I am.

[@mardhiahmichael3343]

Im crying because i dont understand vector space 😭😭😭😭😭😭. Why do i major in math, i can't even do this simple stuff😭😭😭😭😭

[@Nini-vw2zn]

Bruh, 5 minutes in, and I've already gained a clearer understanding of vector spaces than I ever did from attending a lecture and reading the syllabus

[@andrewburnette9019]

So far the best explanation on proving vector spaces.

[@aasutoshthapa2820]

Hey hey. I wanted to ask, does this cover the entire content of the chapter? I haven't watched my class lectures

[@AsparagusSucks]

TYSM this is what I needed. A visual and slow explanation. My UNI just rushed through the basic concepts leaving me super confused.

[@JosePablo24]

simple and intuitive explanation, thanks a lot.

[@meensbeans_]

Thanks for this, I just started Vector Spaces and since I haven’t seen any examples of questions I wasn’t sure how it’s tested. I’m assuming the whole section is tested by proofs.

[@vivekgupta-ib1rd]

if you could relate the topic with physical significance than that will be really great, again thank you very much for making video on these topics!!

[@lalith_kumar_akhila2411]

Excellent Explanation is an understatement..no words...thanks a ton

[@amanuelkassa6719]

This is a great explanation. Thanks.

[@saichaithrik7134]

One of the best explanation ever

[@wontonzz1083]

This is instruction is so much easier to understand comparing to my prof ! ! !

[@watashiwa_rin]

very simple and clear explanation!

[@sharonbell9047]

a very difficult topic for me became clear, thank you!

[@chillsway9550]

Thank you for saving my final exam with this now I can get an A

[@rael213rd]

Did you get an A?

[@priskon5950]

@@rael213rd guess not.

[@dineoramokhothoane6388]

Great video thanks alot

[@phee5192]

thank you man

[@estebansanchezsevilla1930]

great video, thanks!

[@sulgunrejepova3867]

Great explanation Sir big thanks

[@ryantr9832]

this is awesome

[@abubakarseyruu7175]

very satisfied lessons

[@Nohope__]

Professor Strang's linear algebra teaches you , 3Blue1Brown essence of linear algebra shows you, and then this channel saves you.

[@simonelgarrad]

At 6:27,what were you saying 1 to the right 2 up and 3 to the left and 1 down? what does this mean?

[@harineupane4533]

Thank you from nepal

[@sedakun6194]

thank you for good lecture

[@salihsendil]

You saved my life

[@mahimapareek1301]

Please share the link of your previous videos where you proved all these.

[@jyc1815]

thank u for such a good lecture. i have a quick question, in polynomials, what if scalar is zero? is it still in the vector?

[@koreshura641]

very late reply, but if you multiply by the scalar zero you get a 0vector, which is a vector

[@johnaiii1281]

starting at 7:48 I don't get it, how do I turn from polynomial to a vector(with corresponding value for each coordinate or each dimension value)? 'cause vectors has a n x 1 size(n rows, 1 column). pls help

[@rashidminhas6372]

helpull . thanks

[@kathyw.854]

You can't have a non-integer exponent on the variable, in a polynomial. I think a zero degree polynomial would just mean the exponent was 0, so it could be a constant other than 0.

[@althafshameel4466]

Kathy W. yup. I was about to say that. that is not a polynomial in which 3.5 is in power..

[@chiragsharma6215]

@@althafshameel4466 This same thing came in my mind too

[@saichaithrik7134]

Thanks you sir you are the best

[@mountaintiger263]

Very helpful

[@bernardopankaarchegas8007]

Amazing vid

[@shivangvijay9385]

R^2 is a vector space over complex?

[@lm58142]

Great video. Just one small note if I may - the non-integer (7/2) exponent in the example at 08:40 makes it a non-polynomial.

[@pabloandres8757]

I wanted to know why it would make it a non-polynomial

[@daviduzumaki]

because by definition of what a polynomial is, polynomials are not allowed to have non integer exponents@@pabloandres8757

[@vivekgupta-ib1rd]

Thank you for making videos on LINEAR ALGEBRA.

[@kasamadhu3509]

Nice explanation

[@taeh6752]

Thanks

[@scottmoerschbacher8664]

I'd recommend referring to the candidate sets as "sets" or "candidate sets" and not "vector spaces," as a couple of times you said "the vector space is NOT a vector space." Potentially confusing for learners. Best wishes.

[@eku333]

Whoever defined these is a bastard... they're just so confusing for newbies

[@BZ-pv1xd]

and the fact that V can only be a vector space if it's a non-empty set

[@learningacademy7534]

Very helpfull

[@tarkarslanoglu3690]

What if you scale your p(t) by zero. doesn't that make the degree zero?(for the last statement you proved!) Thank you for lectures btw.

[@SuongThol]

good video thanks

[@mbuaoscar3117]

Please I want a tutorial session

[@happygohil6551]

here set of polynomial of degree

[@alex-dn9to]

helll yeah dub.. thx Big Chrev

[@hyderfocus6774]

For the problem at 6:00, saying that if u=(-1,2) and (-1)(2) is not >= 0, this would be false right? As we cannot use a vector that is itself not in the W set?

[@allipraveenkumar1107]

The value of the "x" and "y" has no restriction in the givenvector space,the only restriction is that their product must be >0,so "-2*1=-2" which is kot greater than 0,so the result confirms that it's not a vector space. If you are wondering about(-3,-1) still the product of them will be 3>0 so holds on the given vector space as the condition is xy>0. Hope it helped.

[@Christian-mn8dh]

11:03 how is closure under addition satisfied? what if u + v = p(t) + (-p(t)) ?? that would be the 0 polynomial. confused

[@jaideepkumarpm4769]

Is it necessary that addition is defined for all elements of vector space..or it is enough that the sum of defined elements exist within vector space. Consider Set of all matrixes(including different orders)

[@exol_ruchan]

8:48 how is it a polynomial when the degree is a fraction (7/2)?

[@gradf8678]

8:59 the polynomial in yellow is actually not a polynomial. 7/2 in power means it involves radicals.

[@awaisnaeem4696]

u r right

[@Os_Bosniak]

Degree of polynomial must be in N. 7/2 is not alowed to be a polynomial degree.

[@althafshameel4466]

Osman Delic.. yup I was about to say that. that is not a polynomial..

[@alaska9ine934]

is a polynomial of degree 2 a vector space?

[@hahahahahohohoho5085]

But what if you multiply two polynomials of n degree? You end up outside the vector field. Also, what would be the 0 vector in this case?

[@hahahahahohohoho5085]

@Aenon Solomon you're right. Multiplication isnt an axiom, I don't remember why I asked in the first place. I probably got confused with c multiplication. Thanks

[@edwardpanzi4196]

for 10:45, how can you add two vectors with different dimensions, they must have the same degree.

[@asafahmadshayaan3317]

if a polynomial of degree 2 and we multiply it by a negative number then the whole graph turns upside down .. How is that a vector space then?

[@OswaldChisala]

Hi Asaf! Tl;DR: There is no Tl;DR. Here's my take on your question. Feel free to counter-check its validity. To state things differently, the statement u + v E V stipulates that the sum of u and v is, itself, a member of the group V, on which u and v are properly defined. That being said, the definition of the elements/mathematical objects that V described is left open-ended (abstract). So, yes, if you attempted to define a vector space V = [x y]' : {x > 0, y > 0} multiplying any element of the set by a negative, real-valued constant would, in-fact, collapse the definition of YOUR vector space (the one where we said every element needs to be real-valued and positive). However, if we replaced the condition {x > 0, x > 0} with {sgn(x*y)(x*y) > 0}, your vector space would hold. In the case of a vector space of second degree polynomials, our definition of the vector space, V, might be: V = (a+b*t+c*t^2), for {{a,b,c E R} E I}. Well, what happens if you multiply a+b*t+c*t^2 by -1? You get -a-b*t-c*t^2, which is also a second order polynomial. To see this better, let {A = -a, B = -b, C = -c} and you will see immediately that the general form of the polynomial, A+B*t+C*t^2, is retained. IF our definition said that the coefficients had certain restrictions, for example, a > 0, multiplying by -1 would cause our definition of the vector space of our second order polynomial to fail. So, you see, whether a given general mathematical object (a set of numbers, polynomials, functions, matrices, vectors, etc) can be considered a vector space is dependent on the very precise definition of V for which your object is described. As a closing remark, the term "vector space" can create a false impression of what a vector space is (I hated having to write that, sorry). Why the false impression? A vector space is essentially a parametrized mathematical structure. That viewpoint opens up our sense of vector spaces to consider some of the spaces I alluded to earlier (which are all, in some sense, parametrized), beyond the standard "2 and 3-D vectors" we see in, say, kinematics. This is my present understanding of vector spaces (thanks KZbin!), and I apologize for any misconceptions I may have generated. Best wishes with your study of abstract linear algebra! Yours, Oswald K. Chisala, II The Wishful Thinker :) Michigan State University -- (I'm definitely not a mathematician).

[@aku7598]

If bn =-an?

[@shreyywadhwa]

@8:58 Degree of a polynomial is always a positive integer, it cannot be in fraction. So that example is not a polynomial.

[@thesorcererwiththepointyhat]

also the polynomial with zero degree is a non-zero real number ....the degree of the zero polynomial is undefined (or negative)

[@nofkool]

what is this software ? Great video btw

[@jaysmith4302]

In your definition of V, why is it necessary to restrict c and d to the real number line? It seems like all those properties would hold just as well if c and d were complex.

[@brandonlee2435]

conceptually because no imaginary numbers exist on the xyz plane period, so it could never be within V a vector space consisting of R all real numbers

[@allipraveenkumar1107]

Actually they can be complex numbers or rational numbers too.usually we prefer to define them.a real number line,but we have some vector spaces with scalars as complex numbers or rational numbers or just any field.

[@azultarmizi]

could be better if you have a pointer on which part of the slide you're talking

[@nitink9879]

Do polynomials have directions?

[@cmp95]

good

[@misscocobellemamabolo7133]

Please help me out because I'm a whole idiot. How do I answer a question such as this? Do I draw diagrams? Do choose two random x and y co-ords that contradict the vector space? What do I write in a test or exam?

[@honestdudeguru]

How do I chat with you one on one?

[@Saifullah-oe3gu]

(Y)

[@alanfate4853]

What if V is negative? then u + v can be anywhere. ??

[@alanfate4853]

So that it is not valid even in addition. Am I wrong? I am learning this.

[@shrouqmakki8138]

should we draw?

[@user-bz6nr6ge3s]

RANK OF A MATRIX IS THE SAME AS THE DIMENSION OF ITS RANGE SPACE TRUE OR FALSE

[@AfrikaiMalac2]

I hate having to study this as a business administration major. If I wanted a manager in my company I wouldn't care about his algebra knowledge

[@janecheng9698]

I was focusing on the content of your video until at almost the end of it, i realized that you got a hooooot voice😆

[@janecheng9698]

ohhh great explanations too of course

[@sSSspidermanNNn]

If two or three axioms are good enough to proof things, why we need to have ten?

[@webedblood3994]

because all ten axioms must be true in order to have a vector space that makes sense

[@idrissberchil25]

I am still confused, im still stuck in the physics definition of a vector.

[@danianaj4883]

same bro. same

[@debayondharchowdhury2680]

This is generalization of that physics vectors. These are basically the same vectors that you studied there but you didn't had to study their mathematical properties.

[@danianaj4883]

I received my final score in linear algebra yesterday, and to my surprise, I got 96!!! I was struggling to understand the lessons material, especially in the middle of this pandemic, and changing to virtual classes. To anyone reading this comment and facing some difficulties in linear algebra, believe me, you can get a high score in this subject with helping of youtube.

[@standowner6979]

@@danianaj4883 Good for you! Next up,...

[@aydaasyra7926]

how to solve this question : let V be the set of real numbers of the form (u1 , 1) + (v1 , 1) = (u1 + v1 , 1) and k (u1 , 1) = ( k^2 u , 1). i) is k (u + v) = ku + vu for any u and v in V and any real number k?

[@rameshmahato389]

Its helps me too

[@allipraveenkumar1107]

Is the question accurate??😳

[@secretnobody6460]

-3 and -1 are greater than 0?????????????????

[@anirudhsanthosh2532]

You sound like John krasinski!

[@Laeils]

I don't get it

[@edmondmalepane3141]

I still don't get it … this is difficult for me

[@sheralikhtk5200]

same as me bro.. i skipped vectors chapter in high school and now i regret it :(

[@asdfhjkl5086]

Add the Arabic translation please to understand the details more

[@nathanholbrook1693]

"If you've never looked at a polynomial before.".... c'mon dude. Know your audience.

[@rumanaislam4758]

That's hilarious 😂

[@omerfeyyazselcuk7325]

huh, i thought all exponents in a polynomial needed to be integers.

[@erikkirakosyan5838]

I thought degree of zero polynomial was undefined

[@gunaymirzeli5436]

I don’t understand nothing ☹️

[@woodenstick3517]

I can't understand this math .

[@AymanElKazzaz]

kes emmak shu bet sa3eb el 2eshya

[@muhammadsaad3421]

who tell u to teach? you confuse me more than chat gpt

[@amanuelkassa6719]

This is a great explanation. Thanks.

[@instaminox]

Great video thanks

[@TheDillon8825]

Thanks

[@cmp95]

good

[@cmp95]

good