The Vector Space of Polynomials: Span, Linear Independence, and Basis
We normally think of vectors as little arrows in space. We add them, we multiply them by scalars, and we have built up an entire theory of linear algebra around these objects. However, it turns out that polynomials of degree less than or equal to n ALSO form a so called vector space; that is, they also have a vector addition and a scalar multiplication that obeys the same long list of algebraic rules as vector. This means we can do linear algebra on these polynomial vector spaces in much the same way. In this video we'll cover the standard ideas of span, linear independence, and basis, and see how these ideas from the vector space R^n extend over to the vector space of polynomials of degree less than or equal to n with real coefficients. We'll do a full example showing how to demonstrate these concepts algebraically.
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Пікірлер: 73
I took Linear Algebra in my 2nd year Electrical Engineering undergrad 34 years ago. Now I am watching your videos and doing problems with ease. Too bad KZread wasn't around in 1987. Thank you Dr. Bazette.
@brandonnyamadzawo1452
Жыл бұрын
Kkkkk
@aezazi
Жыл бұрын
Industrial and systems engineering in 1986. Feel exactly as you. Learning/relearning this material is so much easier now. So many great resources at your fingertips. Back then we had essentially one instructor and one or two books.
@manasandmohit
Жыл бұрын
Well well I am currently in EE sem 2
I used the videos from this channel to fully understand concepts from my calc 3 class and I passed my calc 3 class with a 100%. And now I am using this for linear algebra. Dr. Trefor, thank you so much for explaining things in a visual and simple way for us to understand.
@DrTrefor
Жыл бұрын
Amazing!
This is the kind of work that KZread should also sponsor for ever! Mr. Bazett is making a significant contribution to humanity.
Hi, I just wanted to say your Linear Algebra videos are a fantastic resource to learn, especially while we are all remote. Your visuals help me understand the Vector space topics so much better. Thanks
Why your videos haven't got million views yet? It's just so good, so clear and straight to the point. Thank you Doc!
The best part about your teaching is the fact that we can think this intuitively and also visualize the application of linear algebra. I had matrix operations in my high school but never really understood the motive behind studying it. Its through you that I have come to understand the importance of the same.
Mi hermano, this is a real job, eh? A real wonderful job. You really rock, Mr. Bazett!!
Super example. This was where a light turned on for me back when I was taking math.
Thank you Dr, your explanation is so clear
Great material, and awesomely presented , thank you
thank you thank you thank you!!!!!!!!!!! Absolute life saver I finally understand this chapter
Could you do a video about Clifford Algebra (Geometric Algebra) ? There are a few math videos out there explaining the algebra, but I'd love to know how to appy it to real life (engineering) problems. I love all your other courses btw. I was never good at math. I had terrible teachers, I didn't give as much effort as I should, and never managed to get all the rules into my head (well not for the "long term" at least... maily because of a lack of practice) But now watching all your calculus and LA classes and trying to apply your learning strategy and I really feel like I am getting more confident in my problem solving. I set up an anki deck to engrave the rules into my brain once and for all. I am really aiming for a more thourough understanding now. Your videos help a lot. Thank you!
Amazing videos Great work Keep it up Simply awesome
I'm just a french guy who likes mathematics. I really enjoy watching your videos. Thank you for your excellent work :)
@DrTrefor
3 жыл бұрын
Glad you like them!
youre killin it
Wow. Tnx aLot prof. Nice nd simple xplain.
Very insightful!
As a 3rd year math student, who studied systematically most of the topics that you covered in your videos, I can say that i still learnt so many new things, but mostly filled "lack of intuition" gaps in my knowledge. I am glad someone like you exists on KZread. If I knew for this channel 3 years before, I would have gone through much less pain. Greetings from University of Belgrade! P. S. I think you should change your microphone position, you sound kind of shallow. Maybe it's just me.
@navjotsingh2251
2 жыл бұрын
It's all to do with who is teaching you. If you have a bad teacher, they will show you the bare minimum and nothing else.
@thebreath6159
Жыл бұрын
If you want to know more about this read Linear Algebra done right.
Thanks for the explanation
Very helpful!
Really excellent explanation!!!
@DrTrefor
3 жыл бұрын
Thank you!!
shout out to you my man you clutchin up like kyrie in game 7
1000th like woo. By the way I am just curious, is there a case where the polynomials are linearly independent but don't span full P2? other than the obvious case that we don't have a polynomial representing x^2 or x or x^0.
Thank you very much
i love this man
Does a polynomial vector space have to reduce to the identity matrix in order to span a space?
Hey can you make videos on solving differential equations for epidemic models?
Thanks!
Hi! Could we use wronskian to conclude linear independence in this case?
Wonderful
جزيل الشكر دكتور على جهودك في شرح المادة بشكل مبسط ❤ تحية لك من العراق
Thanks alot!
@DrTrefor
3 жыл бұрын
You're welcome!
Really helpful 👏👏
@DrTrefor
2 жыл бұрын
Glad it was helpful!
hello, a question: at 2.30 minutes I can see (c+d)*vecU = c*vecU+d*vecV, but the last seems to be d*vecU ?? Thanks for you reply :)
Great teaching sir what is your country
Thanks
@DrTrefor
3 ай бұрын
Thank you so much!!!
holy damn im week 2 into my linear algebra course and watching this video hurts my brain
what about the span of these functions 1, x, cosx, sinx?
If we take u1=x and u2=-x then u1 + u2= 0 , it is zero polynomial. And degree of zero polynomial is not defined, so, 0 will not be in this space. So how this space becomes vector space?! Please correct me if I am wrong..
at 11:03 shouldn't we have t_1 to t_(n+1) as the co-efficients
Th only reason I could keep up with you is because I drowned myself in linear algebra the past month. Very informative!
2:16 why isn't commutative multiplication a property of vector spaces?
Hello!!! I think , and i say i THINK (i could be wrong), around the minute 11' , wouldn't it be the formula t1.1 + t2.x + ... + tn.x^n-1 ? Or am i making and error. I love your videos by the way ♥ ♥
@julioquejidobarrera8539
Жыл бұрын
Forget about it, i think i understood the meaning
6:03 why would the coefficients need to be 0?
👍
hi, have you learnt machine learning .... can you make a course on math of machine learning??
@navjotsingh2251
Жыл бұрын
Math of machine learning is just calculus, linear algebra and statistics 😀 you may also benefit from learning how to do these topics with a programming language.
There is also scalar's inside the factorization. I seem to be seeing it everywhere now that I know where to look. The scalars are the zeros only. It doesn't scale the regular function. (x-sqrt(2)*r1_He[5])(x-sqrt(2)*r2_He[5])(x-sqrt(2)*r3_He[3])(x-sqrt(2)*r4_He[3])(x-sqrt(2)*r5_He[3])
Oooohh it is so sad that the video has such few views
thank you, you just literally saved my dumbass
funny how right at the end you say its a n+1 * n+1 matrix, but you count the t_n from 1 to n instead of 0 to n
Should "Span {v1,...,v2}" denoted as "Span {v1,...,vn}"? And that "Span {v1,...,v2} = V" denoted as "Span {v1,...,vn} belongs to (not equal) vector space V" ?
Another condition beside not all t1...tk are zero is that all vectors in vector space are NOT zero vectors.
Sorry sir reduce speed of explanation because am not understand
I thought linear algebra only involves linear equations. But polynomial like x square is not linear.
@DrTrefor
3 жыл бұрын
The polynomials are like vectors, and then it is linear in those vectors. Change of perspective.
@nehalamba2747
Жыл бұрын
It deals with linear combinations of vectors and that vector can be a lot of things
Your microphone game is painful here. Otherwise good stuff.
Thanks!
@DrTrefor
3 ай бұрын
You're amazing, thank you!