Introducing Linear Combinations & Span

We saw Vector Addition & Scalar Multiplication in 1.3 Part I. Now we take arbitrary combinations of those two arbitrations, called Linear Combinations. We can compute this algebraically and visualize it geometrically. The set of ALL linear combinations is called the span. In some cases this is set is everything, sometimes just a line, sometimes even just the single zero vector! Two vectors are particularly nice, called the standard basis vectors whose span is immediately able to be determined.

Now it's your turn:
1) Summarize the big idea of this video in your own words
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?
4) Can you come up with your own sample test problem on this material? Solve it!

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.

This video is part of a Linear Algebra course taught by Dr. Trefor Bazett at the University of Cincinnati.

BECOME A MEMBER:

MATH BOOKS & MERCH I LOVE:

[kartikverma]

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

[dennismirante]

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!

[amaanakhtar]

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!

[sharan]

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.

[vaibhavkumar]

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.

[jubuchef]

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

[huh]

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

[Drawtoon]

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

[BingoGoSpace]

Love your course. Legends like you saves us all.

[SS-ybqd]

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

[Bhaumikpk]

Thanks for the beautiful lecture. Back to basics.

[tasninnewaz]

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

[mojtabavahdati]

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

[kathirs]

Awesome videos. Definitely a good LA refresher

[shafiqshams]

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

[kuladeepm.]

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

[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....

[Ravi_Raj_ISM]

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

[foundityes]

04:32 get to the Point :D thanks from germany <3

[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 ?