Dearest Gil, It is beautiful and wonderful to watch you work. You are truly a gifted expositor in Mathematics. You bring richness to students of all ages. John M.

Every lecture should have a "the big picture" part as this one has. Unfortunately in many cases the lectures are done just to do lectures, not to see the "big picture".

It's kind of amazing that there are so many professors out there that don't succeed in explaining this to people, but someone that watched this video could take the information learned and explain it to a fellow student successfully in a short amount of time. I will admit it is possible that youtube videos such as these are doing no more than filling in gaps of professors, instead of informing the student 100 percent more than they were before they watched it.

Watch the TA video on Ax=0 to understand the perpendicular concept. https://youtu.be/LttE1vDVrm0?t=5m29s Very helpful diagram at 5:29. Before seeing that, I didn't really understand the concept of null space and plane.

So the Big Picture (in technical language) seems to be: 1) the row space is the image of the matrix, 2) the null space is the kernel of the matrix, 3) the column space is the dual of the row space, 4) the left-null space is the dual of the null space

3:50 whats happening, "one of the third one of the second" and so on, how can you make a 3d vector like that.. how can you multiply a 3d vector to a 2d plane like that?

for a foreigner, if I understood well, the 'left' means, remaining ? right or false ? but 'remaining' takes much more place to write on the blackboard, that might be the reason of using the vocable 'left'

I was studying for my molecular genetic exam and some how ended up here. It started off well then I was lost when he started talking about " Plain ", first thing pops up in my head is Boeing aircraft plane. This comment is pointless but thank-you for reading.

so various combinations of A are basically vector points that essentially make up two linear lines (because 2 vectors are given here) on a plane. But deriving the null vector N(A) (the perpendicular line to the A space plane) gives you every other possible vector that can sit on the plane. Elegant! So increasing the vector dimensions still allows this method to be used to calculate solutions for multidimensional spaces and I guess N(A) may have multiple solutions that satisfy N(A) thus the solution space may be a line (only one null space vector is possible) or a higher dimensions (more than one N(A) solution). Of course everything has to be calculated relative to the zero origin (because a line or plane does have infinite perpendicular lines/solutions otherwise). So the whole point of this is to quickly be able to determine if a solution (ie multidimensional vector) is in a particular space or not. It seems interesting for applied problems (eg AI, perhaps graphics related) but I feel LA may reach its limits in certain applications simply because it is fundamentally 'linear' (vector based) after all.

Is there a reason why the null space is perpendicular to the row space? 5:23 The explanation only proves that it is perpendicular but not why it would be in the first place.

The left null space is explained using the wrong diagram, it would be much clearer if the professor draw a new diagram for the transposed matrix instead of re-using the column space diagram for the regular matrix

Not quite what I expected from the title, but he is great! Watching the video(s) is very entertaining and informative. Thank you!

He is a wonderful gentleman and a great prof.

Wow… it is so simple.

Dearest Gil, It is beautiful and wonderful to watch you work. You are truly a gifted expositor in Mathematics. You bring richness to students of all ages. John M.

Wish I had seen this when I was taking Linear Algebra. Wonderful short lecture.

sir Gilbert strang

3d classes better

Prof Strang just rock..

Chalkboard Ninja!!

best part at 04:40

"not very thick, is it?

because it's just a line!"

x'D

for everyone looking for even bigger and easier to digest picture https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

Thank you!

thank you professor Strang

Every lecture should have a "the big picture" part as this one has. Unfortunately in many cases the lectures are done just to do lectures, not to see the "big picture".

AWESOME XDDDD

from OCW 18.06, glad to see Professor Strang again.

I love Gilbert Strang's commitment!

confusions untangled with simplicity. Thank you so much. You are a gift to humanity

I want him to adopt me.

good

the legendary Gilbert Strang yeah…

Dude this was good, keep it up, I love how all the concepts of linear algebra just fit together so beautifully

Thank you, prof Strang. Really hope to have you in our school to give a lecture on Linear Algebra.

love it

Brilliant overview of at least half of intro to linear algebra (18.06).

listen with the speed at 1.25 and thank me later.

What did he mean by a perpendicular line is an object in ONE dimensional space? Wouldn't it be two dimensional? Height and width?

is row space also called the Span {(1, 2, 3), (4, 5, 6)}?

So much information in so few words 🙂

So what does the null space relate to in 3 or 4 dimensional space? Or does it have a meaning?

It's kind of amazing that there are so many professors out there that don't succeed in explaining this to people, but someone that watched this video could take the information learned and explain it to a fellow student successfully in a short amount of time. I will admit it is possible that youtube videos such as these are doing no more than filling in gaps of professors, instead of informing the student 100 percent more than they were before they watched it.

Best book I have on Linear Algebra is by Mr. Strang. Well worth the read!

Watch the TA video on Ax=0 to understand the perpendicular concept. https://youtu.be/LttE1vDVrm0?t=5m29s

Very helpful diagram at 5:29. Before seeing that, I didn't really understand the concept of null space and plane.

So the Big Picture (in technical language) seems to be: 1) the row space is the image of the matrix, 2) the null space is the kernel of the matrix, 3) the column space is the dual of the row space, 4) the left-null space is the dual of the null space

Thank you, Gilbert Strang <3

If it wasn't for attendance i would stay home and watch this instead of my classes.

3:50 whats happening, "one of the third one of the second" and so on, how can you make a 3d vector like that.. how can you multiply a 3d vector to a 2d plane like that?

for a foreigner, if I understood well, the 'left' means, remaining ? right or false ?

but 'remaining' takes much more place to write on the blackboard, that might be the reason of using the vocable 'left'

I think this man likes me… he keeps winking at me

Officially mind blown with these patterns!!! How did I not observe any of those!! Need to improve observation skills

isnt [0 0] a 2d zero vector, why is it written in that 3d space?

once again, your teaching is wonderful~thank you

At the end of the video, shouldn't left null space go together with row space and null space with column space?

Thanks! This is really a great video :))

I was studying for my molecular genetic exam and some how ended up here. It started off well then I was lost when he started talking about " Plain ", first thing pops up in my head is Boeing aircraft plane. This comment is pointless but thank-you for reading.

In our class, we call the left-null space just 'null space'. I don't think I have seen left null space

Maybe the people at MIT should invent a blackboard that can fit an infinite plane?

When you didn't study for your final exam and you have 16 minutes left

Great teacher!!! Explained it perfect!!

Yaşlanmışsın usta

what a beautiful metasummary of linear algebra, kudos to prof. Gilbert for the amazing mini lecture.

Thank you for clearing up the concepts that our lecturers messed up!

this is whom I call a good teacher… brief, thorough and "BIG PICTURE" indeed

Does anyone have a link to the previous lecture he refers to right at the start?

wow what a mindblowing lecture deliber. Mr. Gillbert May Allah give you a limitless life for the world.

When you go with a 1.5 OR 2 speed, dear Prof. Gil is quick and vigorous like a young man! It's the fascinating part of online learning.

so various combinations of A are basically vector points that essentially make up two linear lines (because 2 vectors are given here) on a plane. But deriving the null vector N(A) (the perpendicular line to the A space plane) gives you every other possible vector that can sit on the plane. Elegant! So increasing the vector dimensions still allows this method to be used to calculate solutions for multidimensional spaces and I guess N(A) may have multiple solutions that satisfy N(A) thus the solution space may be a line (only one null space vector is possible) or a higher dimensions (more than one N(A) solution). Of course everything has to be calculated relative to the zero origin (because a line or plane does have infinite perpendicular lines/solutions otherwise). So the whole point of this is to quickly be able to determine if a solution (ie multidimensional vector) is in a particular space or not. It seems interesting for applied problems (eg AI, perhaps graphics related) but I feel LA may reach its limits in certain applications simply because it is fundamentally 'linear' (vector based) after all.

This man will always hold a special place in my heart

Thanx a lot

thank you for a clear vision on the big picture of linear algebra

Hi, is that for first year of Maths at a university ?

That was poetic.

Thank you so much sir! I enjoyed this beauty of math while I learned via your wonderful teaching! God bless you!

I have an exam in 20 minutes and I'm watching this at 2x speed

I feel special when he winks at me. 😉

Beautiful lecture!

Great video thanks!

Simply super i bcme fan of u sir

i began to find Linear Algebra an interesting Subject after watching this open course!

Okay, where do look for videos that start from something more like "2+2=4"..?

really good teacher but what about linear transportation , determinant etc

u are great. thank u!

If someone knows could tell me what is the course book in which lessons are based on? I'll be thankful

I want to learn this but have no mathematical background, where should I start?

Me after watching this video : So what are matrixes??

These subspace are

THICCreally great! but go with speed 1.25

Insightful: row space is perpendicular to null space, so is column space to left null space.

An amazing teacher!

Me after watching half of this video:

I'm smart…..

Not

MiT is MiT :V

15:51. No: thank YOU, Sir!

I have Gil's latest book ("Linear Algebra and Learning From Data"). It's just like listening to him talk.

I love this Professor. He is amazing and I'm really grateful that he did this , and grateful that MIT hosts it. Thank you

i wish i could understand this, guess some brains just cant click like others…

Is there a reason why the null space is perpendicular to the row space? 5:23 The explanation only proves that it is perpendicular but not why it would be in the first place.

curse this subject. curse it…

of the 4 sub-spaces, "left nullspace" …really? There's gotta be a better name than that we could use?!?!

Implies that although, in an nXm matrix, n may not equal m, the number of

independentrows always equals the number of independent columns.I love you ❤️

Keep the good stuff, precious you

IS THIS FUNDAMENTAL THEOREM OF LINEAR ALGEBRA??

At last. A clear picture. Thank you professor.

I love this guy. Best math teacher ever

The left null space is explained using the wrong diagram, it would be much clearer if the professor draw a new diagram for the transposed matrix instead of re-using the column space diagram for the regular matrix

Big, no – great teacher!