Linear Independence of Functions & The Wronskian

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What is the analog of Linear Independence for functions? If this was just vectors linear independence would mean the only way you can take a linear combination that adds up to zero is the trivial linear combination with all the coefficients being zero. Same for functions. However, we do have one extra tool called the Wronskian, which is a nifty little determinant that we can use to create a quick test for linear dependence of a set of functions. The connection to differential equations will be that we will be looking for a fundamental set of solutions which will have to be linearly independent, more on that in the next video.

0:00 2D case
1:22 Geometric Picture
2:27 Linear Independence
5:54 The Wronskian
8:37 Example

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Dr. on 0:48 aren't you saying the opposite of what the text states? The text being the correct one in this case right?

joshuaisemperor

Sir, I had to discontinue maths in college due to financial reasons, but I find it fascinating, and I have decided to start learning maths and physics from where I left it in college. I am so thankful to you to provide such contact free of cost. a lot of people don't tell you how help full these courses, I absolutely love you and please continue to teach and spread your knowledge free of cost like this. I am here only coz I am interested in the subject and i can't even tell you how happy i am to have found your channel. Thanks a lot for everything sir.

aakashdharmakari

I really love how you explained the Wronskian clearly.

ogunsadebenjaminadeiyin

I am doing ODE as a part of my university math course.Worked on linear algebra last year (from 3blue1brown). Seeing same thing from two different perspectives and connecting the theory behind is something I like about Math most.
Getting comfortable with ODE. Maybe will work on it even after my semester.
Thank you SIR.

uthsoroy

Doing Differentials, and I got to admit you helped me out through with your videos. Sometimes, I second guess myself and often cases concept grasping is difficult for me. Now I am doing better, thank you sir!

DaGitarReaper

I love your videos, they are easy to follow, they make sense and it all comes so nicely together!

imaginary

Professor Bazett, this is an exceptional video/lecture on Linear Independence and The Wronskian. These are two powerful tools from Linear Algebra that is also used in Differential Equations.

georgesadler

I was struggling with my university lectures, thank you very much sir, Nice presentation

jonsnow

great. keep it up. keep educating the students who are sleeping in their maths class.

anchi

great video, really cleared things up

cesarmorenoy

I think you're erecting a large, fictional wall between functions and linear algebra. Function spaces ARE vector spaces when equipped with appropriate, obvious operations (although often infinite dimensional), and studying them this way IS linear algebra.

MasterHigure

great video, much better than my professors indecipherable notes :)

rudyj

What a great mathematician!!! U helped me and my frineds a lot...
Thank you!

Reptilian.cricket

The Wronskian? More like "Sharing great information is your mission!" 👍

PunmasterSTP

Laplace vids would be rlly helpful rn my semester ends may 1

trexbattle

You said at the beginning of this video that 'sin x and cosine x are NOT linearly independent' probably a slip of tongue. Actually they ARE linearly independent, as you go on with the counter example.
I am watching the video now- which is your latest one. You may correct the inadvertent error. Or, correct me if I am wrong!

utuberaj

My book diff. eqs. for dummies says the wronskian is a determinant that gives you the constants of the functions.

the_eternal_student

I clicked so fast as soon as I saw the notification of your video

arandomghost

I know this is an older video but I was wondering, what would w(t) look like if you had more terms w[y1, y2, y3] would it be something like

zesnych

7:33 why the wronskian = 0 somewhere and not everywhere?

nicolabellemo

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