Null space 3: Relation to linear independence | Vectors and spaces | Linear Algebra | Khan Academy

Understanding how the null space of a matrix relates to the linear independence of its column vectors

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[ozzyfromspace]

Thank you so much! I can imagine most computer programing environments (MATLAB, python, the whole bit) are optimized for row-reduced echelon form. What I couldn't see was why it was even worth the effort to start with, and this is certainly an excellent reason! By generating the rref(Some_Matrix) all it has to do is check for free variables in order to tell you if your vectors are linearly independent or not. I'm very impressed by this line of reasoning. I'm interested in computational mathematics, so seeing ideas for things that make it easier for computers to generate insight is really cool. I wish you the best, Sal! I'm not even taking linear algebra, I just thought it would be a nice thing to learn so I've been watching your playlist from video 1, and you've made things so very easy to understand. Thank you!

[YoutubeFella]

oh my god, SAL, you have perfect timing. I was checking here just a week ago because my linear algebra class is getting tough. thanks so much!

[dfsfklsj]

To make more connections...
1. Matrix A has to be a square matrix if its column vectors are linearly independent.
2. Matrix A reduces to the identity matrix when its column vectors are linearly independent.

Very interesting stuff. Definitely going to think deeper into the subject on my own time.

[alavalasuraj]

AT 8:51, the matrix can be any diagonal matrix, not only identity. The video has very good insights.

[rfmvoers]

8:11 "this is going to be a square matrix" No. As a counter-example, consider the 3×2 linear system 5x+3y=0, 2x+2y=0, 7x−y=0. In a graph, these equations are three lines going through the origin. The lines are not parallel, so they only intersect in the origin; the null space is {0}. And, as you explained, this means that the colums of the matrix must be linearly independent, which is true. But the matrix is not square.

[elai]

Most enlightening 11 min of linear algebra I have ever seen. thank you SO MUCH!
btw. does this mean that any linearly independent matrix that is row reduced will equal its:Identity matrix?.. and that the dim of the NUL of the matrix must always = 0, therefore the dim of Col(A) must equal = the n number of columns which also equals the of rank(A)?

[payamem]

may alimighty lord bless you and your family for great afford of yours

[eongreen]

@tareqhardan if you're talking about the nullspace being the zero vector, then that means that the columns of the original matrix are linearly independant

[ionglacier]

why does it only takes you to perfectly explain an entire lecture worth of information in 11min while my prof takes an hour and a half to explain the same thing which is also not as clear? lol great videos

[lightfireme]

@pedroissler I'm sorry, but I disagree with you. Some teachers explain math in an extremely technical way, making it hard for students that aren't math savvy to understand. And heck, some teachers are just lousy. I think Sal is extremely clear and informative in his delivery.

[futterkulcha]

Why can there not be free variables? If all Xs are zero the free variables would become zero in the end anyway, right?

[xmoelifex]

i jus might nt fail my linear algebra exam should cm teach at the university of toronto. :)

[stevenan]

does this mean that a matrix HAS TO BE SQUARE for it to be linearly independent? hmm.

[charanreddynallapareddy]

Sal, is it neccesary that independent vectors spans everything in subspace

[arthursb]

@pedroissler Man, I have to disagree. I'm not even in college, but I know derivatives, integrals, and now i'm learning linear algebra all thanks to Sal. I think this videos could substitute teachers "in real life".