The Null Space & Column Space of a Matrix | Algebraically & Geometrically

To every matrix, there are two natural subspaces: the Null Space of A, and the Column Space of A, denoted Null(A) and Col(A). Col(A) is the range, or all the vectors "hit" by a transformation. It turns out this is equivalent to the span of the columns of A. Null(A) is all the vectors in the domain that are sent to zero by the transformation. We visualize these two subspaces, prove they actually are subspaces, and show how to compute them.
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[@alkankondo89]

Your lectures are very complete and thorough, especially with the supplementary visuals you include! This particular lecture was a great review for me for a lot of the abstract concepts of Linear Algebra that I was fuzzy on. With all the pausing and mental reviewing I was doing, it took me easily half-an-hour to get through this 10-minute video, haha!

[@sollinw]

thought it was only me!

[@stevehof]

When you love a youtube series so much that you watch more than 60 videos in 2 days. I, in fact, tutor linear algebra and this series has completely reshaped how I'll introduce certain concepts, and the order I'll introduce those concepts in. Hands down the best introduction to linear algebra youtube series in existence.

[@manuelkarner8746]

i cant stop binging this series, I get so many dopamine kicks from every aha-moment, like very 30seconds, it´s funny

[@AlePreludioFinal]

These videos are pure gold.

[@veggieveggie1215]

why is this man so good at teaching

[@rotarydev3891]

You are a legend! Seriously your videos help me tremendously and I would struggle far more without you. So thank you very much!!

[@maurocruz1824]

Your particular vision of this stuff is amazing. Is there a geometrical meaning for the row space as well?

[@bill7064]

The visualisation in this video is very helpful. Than you :)

[@shivkarj1456]

Thank you so much for your amazing tutorials. Linear algebra is so interesting when you explain it.

[@DrTrefor]

Glad you like them!

[@nhupham8127]

At 0:29, T(x) = [1 0; 1 0]*x The second column should be (0,1) instead of (1,0). Your video is greatly comprehensive; everything is good except the mistake of the second column I mentioned above.

[@elfabri666]

No, the matrix shown in the video is right. Notice that both columns are the same vector, (1,0), and that is why it spans the x-axis. If we have Ax=b, with x any vector from IR^2, let's say (a,b), a,b real numbers, then Ax=(a+b,0). You'll understand this idea better when you have a notion about a space generated by a set, the rank of a matrix and its dependenciy. Hope it helps :)

[@abdallahalsahhar9511]

amazing lesson, Thanks for time and effort

[@fotisrokanis4919]

. The idea to represent geometrically linear algebra was awesome. Linear algebra is (rightfully) taught in an abstract manner, but usually to first year students who are not used to the abstractness of pure math. Are you going to make a video on quotient vector spaces?

[@fotisrokanis4919]

Trefor Bazett It does. I study mathematics, with most of my courses in pure ones (Next year will be topology, abstract algebra II, differential geometry II ect).If I can help you in any way (prepare notes for a video for example) l'd be glad to do so

[@shafiqshams3065]

This is so amazingly explained but still its really hard to get it completely.

[@DrTrefor]

I agree, this concept really needs you to sit down and wrestle with it for a while and come back to the video after doing that

[@jan-willemreens9010]

...Crystal clear lecture, Dr. Trefor... Thank you very much.

[@AbjSir]

Thanks

[@sergiolucas38]

Nice video, thanks :)

[@DrTrefor]

Glad you liked it!

[@advancedappliedandpuremath]

Hi Sir how can we find the null space of set of vectors from M2x2

[@raginibhayana8305]

how do you plot the matrixes on the graph?

[@Compassrome27]

Thank you so much sir. Saying your work is an amazing piece is just such an understatement. Over here you don't just present mathematical concepts with utmost clarity but also with burning passion. You are a legend 👍

[@christianfernandes2604]

Hi Dr. Bazett, right around 9:04, you talked about the conditions for the Null(A) to be a valid subspace. What if you have the situation where the ONLY solution to Ax=0 is when x must be the zero vector (i.e. A is the identity matrix)? In this case, x in Ax=0 would be zero and there wouldn't exist a y vector such that Ay = 0 (because no other input vector exists to solve for Ay = 0). Can you still make a valid claim that Null(A) IS A SUBSPACE by arguing that if you have Ax = 0 and if we add it to itself, Ax + Ax = 0 such that A(x+x) = 0? I hope my question makes sense.

[@huychaungoc830]

if A is a square matrix, Does Null(A) = Null(A^2)? please help me

[@HeyEveryoneHi]

What the hell? I just watched five videos on linear algebra and now I understand it! Without even taking the linear algebra course

[@HeyEveryoneHi]

@@DrTrefor much love

[@yihongli7252]

你是我的神！！！！！

[@dhananjayjoshi3561]

Will you please explain row space in the same manner

[@dhananjayjoshi3561]

@@DrTrefor thankyou very much. We would love to watch more videos on linear algebra as well as real analysis.

[@gate_dose]

[@GS-ke8ex]

0:39 Im pretty sure one can easily find a vector that does not collapse to the x axis when multiplied by that... Right...?

[@mathrovert]

The b vector (after transforming vector x) would be < x_1 + x_2 , 0 > Your 2nd component is zero.

[@mardzj]

6:58 Subsets of Rm or Rn?

[@mardzj]

I don't understand why in one case it should be n and the other m

[@mardzj]

@@DrTrefor thank you!

[@sollinw]

[@zegervg8152]

merci wi matje ma paul igodt legt et beter ut

[@anilsharma-ev2my]

Whole universe is a matrix amd god is its determination factor which is zero solution for it's original position and equation for systems equation of all types🤓🕉🕉🕉

[@SS-pl7ci]

No enough examples.. very abstract.