Linear Algebra - Lecture 15 - Linear Independence

In this lecture, we learn the definition of linear independence and linear dependence. We work through several examples to illustrate the definitions.

[sankoktas]

Dude is x10 better than my college prof

[micah]

I’ve been teaching myself with your lectures, I was caught off guard when you said many people have trouble with this, it seems very straight forward

[snehatimilsena]

Your explanations are great. I learned in so much short time. Thank you and god bless you for saving our time and lives!!!

[sana]

Your course is awesome and i would say that it really changed my visualization about linear

[saulorocha]

Always direct to the point and clear. Thanks

[christophertech]

Sir God bless you, the way you explain mind-blowing

[LLai-zhbk]

thx for making these awesome videos !!!

[shauntecodner]

I'm mind-blown how easily you explained this! Thank you so much! I have my finals tomorrow and I just found your page. I'm basically learning everything from your videos

[ajwaabid-ngdx]

Thanks for videos and please more lectures provided

[eshuuu]

Hello, your videos are super helpful! Thank you for putting in so much work to create them. By any chance, do you have a video on LU decomposition?

[atodaz]

How would you explain this graphically? If you have linearly dependent vectors, is one not needed to span?

[maxpercer]

If I understood this correctly, given a set of m vectors { a_1, ... a_m} where a_i ∈ R^n, call this set of vectors B, if we apply 'row reduced echelon form' on B, rref(B) for short, then by looking at the number of row pivots and column pivots of rref(B) we can answer if there are any redundant vectors (linear dependence) in B and whether B spans R^n.
1. Does rref(B) have a pivot in every column (i.e. no free variables) ?
Yes - B is linearly independent, there are no redundant vectors.
No - B is linearly dependent, there is at least one redundant vector.
2. Does rref(B) have a pivot in every row (i.e. no row of zeros)?
Yes - B spans R^n. No - B does not span R^n.
Also, note that If you answered yes to both questions then B is a basis for R^n, which in that case, m = n.

[mikesgarage]

Wish I had this in college. My lectures were only proofs and theory. Almost no examples given. Needless to say, I almost failed the course because I failed to understand critical concepts fully. I've been studying Linear Algebra on and off since then... for almost 12 years now. If I knew then what I know now...

[dktchr]

it may be in lectures to come but what is the meaning of these terms, linear independence/dependence and dependence relation. Its clear what they desccribe but its not clear why those terms.

[xoppa]

dont choose x3 = 0 for linear dependence relation. :P

[tecrahmutungulu]

please be slow on elementary matrices so i can follow carefully

[medardoramirez]

What does 0 = 0 tell us? ; 4:50
Do we just say C1 = 0; C2 = 0; C3 = 0.