Introducing Linear Combinations & Span

[@kartikverma1899]

Sir every maths students need the teacher like you . You made my day love you sir. Love from india

[@sharan9993]

The whole idea of lineae combination is explained in one minute. I always come back to this video whenever i lose the concept of what is actually a linear combination. Thank you so much for dumbing down the concept to us mortals.

[@dennismirante9528]

This is a great series! There are many concepts I found hard to understand when I took this course. You've made them clearly understandable!

[@amaanakhtar4115]

For anyone wondering how he got those values at 3:07 We know that there is some value of x and y such that: x(2) + y(-1) = (0) (1) (1) (-2) Now this is nothing but 2x-y=0 x+y=-2 (Just did some scalar multiplication) This is a linear system of equations and now we can continue to find the values of x and y if they exist. (2 -1 | 0) -----------------> (2 -1 | 0) (1 1 | -2) R2 -> R1+R2 (3 0 | -2) Now we can substitute the values. 3x+0y=-2 x=-2/3 2x-y=0 2x=y 2(-2/3)=y -4/3=y And we're done!

[@vaibhavkumar7745]

A few months ago I commented on your video; why your channel has mere 69k subscribers, coz u deserve a lot more... And now I see 112K... I knew it. Good things take time to be successful, and now I see a positive sign. Thanks a lot for making the math so beautiful.

[@DrTrefor]

Thank you so much 😀

[@Tom-ud8js]

and now 317, amazing

[@jubuchef1348]

Daaaammmn, I didn't understand this at the uni lecture, and now I understand from your video!!

[@Drawtoon123]

this video is really helpfuul for me thnk uh sir respect from india

[@naman4067]

Thankyou for thanking a person for entire india you made my work easy . Now I don't need to thank him.

[@huh5950]

This compilation is juts t beautiful, I don't know how to thank you

[@stevehof]

Absolutely outstanding videos!

[@SS-yb1qd]

I loved the explanations. No wonder if u get 1 million subscribers.

[@DrTrefor]

I hope so!

[@BingoGo2Space]

Love your course. Legends like you saves us all.

[@Bhaumikpk]

Thanks for the beautiful lecture. Back to basics.

[@rushalimhaske4450]

Thanks again & again

[@amirhosseinshahbaz9641]

These vids are perfect and admirable.

[@mrtony5778]

Thanks from Republic of Korea~^^

[@webper3630]

Thanks from Uzbekistan

[@mojtabavahdati7648]

My thanks are basis vectors and I give the span of them to you.

[@tasninnewaz6790]

You are cxcellent !!!!! You have no more subscriber but i will suggest this chennel of my math friend. Love from bangladesh.

[@kathirs1]

Awesome videos. Definitely a good LA refresher

[@shafiqshams3065]

Very nice, please make a video on an affine combination, a conical combination, and a convex combination. Thank you :))

[@missoss]

Never stop making videos.

[@asht7788]

Concise and clear. Thank you

[@bextuychiyev7435]

Awesome expanations, thanks!

[@kuladeepm.2655]

Thank you sir , your videos are very helpful :) Can you suggest me any reference book for algebra

[@BiancaAguglia]

In one of the comments for the 7th video in this playlist, Trefor recommends the free and open source textbook joshua.smcvt.edu/linearalgebra/#current_version 🙂

[@kuladeepm.2655]

@@BiancaAguglia Thank you 😁

[@oraange]

Amazing lecture! I have a question ! If I have span of two vectors and I have a vector indepedent from the span of the two vectors. Does it look still as a plane ? If yes, is the plan finite or infinite ?

[@humphrieswilliam9341]

no, if you have two vectors and a vector independent from the span of the two vectors ,then the set of the two things doesn’t lie in a plane .instead they will lie in a 3D space

[@simonpeterj]

Appreciate the effort 👍

[@ItsMeTheUser]

Thank you so much sir!

[@ShubhayanRoyChowdhury]

Q.. How do we decide if a vector is in some span? Ans.. If the vectores are the linear combination of standard basic vector.

[@pranavrocks2002]

Too good sir

[@saurabhsingh-ow7ue]

thank you sir

[@continnum_radhe-radhe]

🔥🔥🔥

[@Ravi_Raj_ISM]

thanks a lot sir. also thanks miss ism who shared this video :)

[@sergiolucas38]

great vid, thanks :)

[@foundityes]

04:32 get to the Point :D thanks from germany

[@ibrahimalotaibi2399]

Informative series Dr. Trefor. Though, you are using too much energy to illustrate without the need to do so. Try to reduce the amount of energy you are using to explain things. It will be beneficial for you & us.

[@kartikeyedunite]

Hi, If Cartesian Coordinate System gives just scalar multiplication to basis vector (i, j). What are polar coordinates doing... In which one in just scalar distance(r) and the other angle is?.. Which type of tensor is angle in the set of (distance, angle, area volume). Could you make sense of this....

[@angelapepito7133]

thanks doc ur a g

[@pinklady7184]

You might want to use stronger, brighter neon font colours against gray/dark backgrounds, as there are some colour-blind viewers. And also night blind viewers. Your red fonts and blue fonts are indistinguishable to my eyes, their hues lost in gray background. Opt for neon colours, that are brighter and more distinguishable.

[@forughghadamyari8281]

Dark blue is not a good color

[@datta230]

i think this video could have been more simple

[@daoudatraore930]

Hey Trefor, Sorry for the silly question but i got a doubt about the the minimum dimension a vector could exist. You've said in the video that Xi belong to R and vectors (ai) belong to Rm. And in your previous videos you started talking about vectors using a plan, which is R2. So my question is can i have a vector in R ? If yes or no please tell me why. Kind regards

[@daoudatraore930]

Hey Trefor, What you've said helped me a lot, and it makes more sense to me know cause if the vectors are in R^n and n > 0 then R^1 is part of R^n. So it's clear for me now. But i would like you to explain what you meant by saying "When we use R^1 we are thinking of it as a vector space in its own right." please. Thank you very very very much for helping me

[@АлексГудев]

​@@daoudatraore930 R is the set of real numbers - you can multiply any n-dimentional vector from Rn with a real number. On the other hand, if you're looking at a line - R1, that is, a one-dimentional vector space, and have fixed a center O, then the vector space R1 consist of single-dimentional vectors, represented with a single number from R, but nevertheless we still think of them as vectors. I think your confusion stems from the fact that we're looking at a single object from two different points of view - once, R is the set of real numbers, and then - a 1D vector space. (1) By the same logic though you can be confused about R^n being the set of ordered n-touples of numbers like (1,...,n), and at the same time - a vector space of n-dimentional vectors. (2) The relation in (1) and (2) is the same - but yet, you're not confused by the latter. Try to see how (1) is the same as (2), accept it and let your confusion vanish.