Determining Linear Independence vs Linear Dependence

Learning Objectives:
1) Given a set of vectors, determine if they are linearly independent or not.

This video is part of a Linear Algebra course taught at the University of Cincinnati.

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[isaackay]

Hey Trefor Bazett, I just wanted to thank you and share my gratitude with you for making this series. This past month I've taken Linear Algebra and C++ simultaneously, while also holding down a part-time job working QA at the SMU Physics Dept; so as you might be imagining, I didn't quite have the luxury of free time to spare. I ended up reviewing for both finals this past weekend, but still struggled to fully understand how everything connected together: I honestly felt like I was completely screwed. However, by way of happenstance, I googled, "Visualizating Diagonalization and Eigenbasis", and came across your video lecturing on it. After watching it and still not grasping the related concepts, I decided to start watching your playlist from episode 12 onward (keep in mind this was at 3AM this morning--my final was at 1PM). I opted to wait and look at my final review after I completed your 81st video that way I could make the most out of my time and go into it knowing what I needed to know. Well, friend...I gotta say, you basically taught me the basic introduction to Linear Algebra in less than 10 hours (I watched most of your videos at 1.5-2x speed with the most of the final videos at 1.25-1.5x speed). Just a few hours ago I came out of my exam, and I finally received my






85%!!!!




Now, I'm slightly disappointed I messed up on a few silly mistakes I made, but dang, man, I have to say I COULD NOT HAVE ACCOMPLISHED THIS WITHOUT YOU. Thank you, sir, for all the time and effort and attention to detail and specific example problems that you put into your videos.

If anyone else is reading this after watching this whole playlist, congratulations! You now have a pretty solid grasp of the concepts of Linear Algebra.

If you're reading this, but have not watched this many of Trefor's videos on Lin. Alg. and are struggling to grasp the concepts, DON'T WORRY! There's a UNIQUE, yet TRIVIAL Soln. (; to your problem (ha..I'm all you need to do is sit down and watch through the majority (if not all) of the videos in his playlist. If you're short on time, that's still no excuse not to go through it. I LITERALLY did it in LESS THAN HALF A DAY, and still had time to eat and review other material. That means you can/should too; you won't regret it.

I'd love to post my two exam reviews along with the blank copies of the two actual exams I took this month, all of which include solutions with explanations of course; however, I'm not sure if I can do that here. If I can't, or you can't seem to find it please don't hesitate to DM me. I'll be more than happy to provide that. (I'm a programmer, so I'm almost always on my computer, and if I coincidentally am not, I most certainly have it nearby, so you can expect a pretty rapid response.) Heck, maybe I just might send it to you, Trefor--that way you can do another couple videos going over them!! That'd be awesome. I'd actually watch you go through them again.

[andrewhartman]

This is the most helpful video I have found over 5 days of searching! Thanks for your work

[jingyiwang]

I have been watching your mathematics tutorial since I started learning calculus in my freshman year. Now I am learning differential equations. I am really grateful for your help along my way.

[malakzeriouh]

I guess there's a mistake over there; when you've replaced R1 by -R1 you should have gotten (1 -3 -5) instead of (1 -3 5)

[hershyfishman]

You don't have to solve the system. You could just look if it has a pivot in every row or not. If it doesn't, then it's linearly dependent. If it does, then it can only have one solution which would be the zero vector.

[bextuychiyev]

Amazing explanation of linear dependence in the last video!

[abhashkumarsingh]

Hello Prof,
Is there an example where some of the ai's get to be 0 for linear dependence?
How to visualize such case?

[piyushmittal]

We can use determinants(THE WRONSKIAN)

[ryanabel]

hey prof, a very interesting video. Actually, i have a question if i have a matrix A to be linearly independent will the eigenvalues of this matrix be linearly independent? thanks in adavance

[chiranjib-konwar]

i am your first viewer. Very well described. Thank you so much.

[hamsack]

I don't think that's right. Wouldn't RREF be ?
1000
0100
0010

[delirium]

is there a way you can do it without finding the row echelon form ?