Vector Spaces | Definition & Examples

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Vectors in R^n obey a list of rules, things like commutivity of vector addition that a+b = b+a as vectors. However, we can abstract this list of rules and introduce the general concept of a vector space. A vector space has elements that might be all kinds of weird things, but as long as they obey the same list of rules, they are a vector space. And the key here is that all the things we have done in linear algebra extend to this new situation too! Our first new example is polynomial of degree less than or equal 2, which have a natural "vector" addition and scalar multiplication.
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2) Write down anything you are unsure about to think about later
3) What questions for the future do you have? Where are we going with this content?
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Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master math means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK.
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This video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.
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Опубликовано:

15 июн 2018

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Комментарии : 60

@mrawesome2791 5 лет назад

That's a great video, learned a lot, you're doing a great job, but could anyone please help me? At 02:32 the second equation in the green group, shouldn't that be: (c+d)u = cu +du

@mrawesome2791 5 лет назад

@@DrTrefor hello sir, it's been a real honor for me that you replied me, love from India!

@naman4067 2 года назад

No its fine

@diegoguisasola3858 3 года назад

You explained in 8 minutes what my college teacher couldn't explain in 1 semester. Excellent work!

@vaibhavjawla6915 2 года назад

Me too

@minma02262 3 года назад

You are like my dream teacher. The straightfoward explaination is extremely relaxing. I'm not scared that a hodgepodge of terms will be thrown to my face. Thank you very very much for making this.

@DrTrefor 3 года назад

Wow, thank you!

@mathadventuress 2 года назад

Yeah and so handsome

@jstadler417 5 лет назад

Thank you for your hard work, Trefor. This video series is very helpful and much appreciated!

@sthabileduma341 2 года назад

Duuuuuude! I'm writing a test in a few minutes and you just saved my life once again.

@rockyjoe3817 5 лет назад

This playlist is amazing , I will watch all of your videos!!!

@chileshechitabanta9995 3 года назад

I appreciate your hard work, always getting to understand more from your explanations

@sohumramouthar9722 3 года назад

This channel is truly underrated.

@AlePreludioFinal 5 лет назад

Best explanation ever!

@gentlemandude1 4 года назад

Amazingly clear explanation!

@gangkilo9859 3 года назад

i like your energy when u teach

@tanjinaaktar1146 6 месяцев назад

One of my favorite teacher ❤

@miklosszolnok4751 4 года назад

Instantly subscribed! Keep up the good work!

@sujitbaruah4536 3 года назад

Excellent teaching

@catcen9631 2 года назад

THANK YOU SO MUCH!

@princeGamer-1 3 месяца назад

After watching the English tutorial, you explained all the key points of vector space linear transformations abbreviated easily.

@hopeless8473 Год назад

I'm from Bangladesh 🇧🇩 Your class was so much helpful. Thank you 🖤

@user-ze2ju3rm7u 2 года назад

Thanks for the clear explanation. Love your energy. And you have beautiful eyes.

@johnlovesmath 2 года назад

Very helpful.

@user-uv5do7nz5o Год назад

Dr . Trefor … thankuh so so so much ❤

@mojtabavahdati7648 3 года назад

You've mentioned at least twice in the video that translation between different vector spaces (specially from Rn to other vector spaces) would be of great use and importance; So I wish you provided an example of such translation as well. But this is very helpful, makes some things feel less abstract to me. Thanks.

@DrTrefor 3 года назад

That's fair. A bit later in the playlist is videos on the polynomial vector space which is our first really new one, and this has a direct translation

@mojtabavahdati7648 3 года назад

@@DrTrefor Great, I had no idea this was part of a playlist, I've found it now. Really cool playlist!

@maazahmedpoke 5 лет назад

are there any videos in this series about the kernel and range of a linear transformation?

@kathirs1 5 лет назад

If you could generate transcripts for these videos, it will be really helpful !!

@amardeep7428 5 лет назад

Nice

@kathirs1 5 лет назад

Hi Professor, if my understanding is right, vectors in R^n (for example, lemme take R^2, in 2 dimensional space) which are a set of vectors which obey these rules like commutative, vector addition and scalar multiplication form a space called vector space??

@Fania973 Год назад

I wonder if you know how many students lives you're saving. THANK YOU

@tasninnewaz6790 6 лет назад

Love from core of my heart

@puturnamehere777 4 года назад

Hi Trefor! Just to reaffirm my understanding, when you defined vector spaces, it is not only defined for vectors (like what we’ve seen so far) but anything that obeys the rules at the beginning of the video (in this case, the example of polynomials)?

@ale-hl8pg 2 года назад

(if anyone still needs the answer to this) I'm pretty sure that vector is used to refer to any element of a given a vector space, for example, a polynomial of degree n could be considered a vector of a vector space of polynomials of degree n (the degree of the vector space could be greater though, a polynomial of degree 3 could be located in a vector space of polynomials of degree 5, similarly to how an arrow vector with a z coordinate of 0 is a vector of 3d space despite seeming like it fits in 2d space, it depends on the context you view it in). Since a vector is just a member of a vector space, that would mean that your assumption is correct as a vector space is any set that follows those rules

@kathirs1 5 лет назад

if any 2 arbitrary vectors in R^2 space obey these 10 vector addition and multiplication rules, is that mean these 2 arbitrary vectors forms a vector space or the span of these 2 vectors as vector space ?

@boyang6301 4 года назад

You voice is very deep, so you'd better use a microphone which can cancel echo. It will be wonderful.

@altuber99_athlete 3 года назад

And talk a bit slower.

@user-ts8rc7cn1r Месяц назад

For me it's fine

@mathadventuress 2 года назад

Can you do a. Video on how 2x2 matrix is a vector space or 4x8 matrix

@AmrouniYahia 3 месяца назад

Excellent content. U just need to improve the audio quality and organise content more maybe in a web site or re arrange videos in playlists in cleaner way

@foziadawood9520 4 месяца назад

Hello sir please recommend a book of linear algebra

@tlotlooepeng8721 11 месяцев назад

bravooo

@InnoSang 3 года назад

3:36 that's the face I make when I try to understand math

@DrTrefor 3 года назад

haha:D

@Polyviani 3 года назад

What’s the point with vector space? How is it different?

@nonhereticalmanofculture5401 3 года назад

Would you kindly add generated subtitles sir? I am not native speaker

@tube7788 3 года назад

Need help s={1,x1,x2,.....,xn} Is this vector space of R³?🙏

@DrTrefor 3 года назад

Doesn't contain the zero vector, so no

@tube7788 3 года назад

@@DrTrefor but x1,x2,...,xn are element of real number

@rhl475 Год назад

I feel you are saying a good things, but can you turn on or add(I don't know) the translation 😚

@Fine_Mouche Год назад

Matrices are a vector space ?

@thisisaman4 4 месяца назад

i came here, to get the feel of vector space, i already know all these rules, everyone is doing it, i just can't visualize what i am writing. That's the problem

@josecanedo007 2 года назад

Why mathematicians don't explain why division is not part of an operation of the vector space?

@naiko1744 Год назад

You can divide by any scalar by multiplying, if you wanna divide a vector by 3 then just multiply it by 1/3. On the other hand "multiplying" a vector by a matrix, while called multiplication, is actually a linear combination. Similarly, "multiplication" of matrices, while called multiplication, is actually a composition of matrices/transformations There are as many divisions as there are multiplications, it's just that they named a lot of different stuff "multiplication" ...