Thank you for the video!! I just learnt about the nullspace, range and rank-nullity theorem in the more general context of linear transformation functions. Before that, we learnt about the rank-nullity theorem in matrices only...I think I finally got the link to the special case where we look only at matrices Our notes defined nullspace as N(T)=the set of all the vectors v in vector space V (the domain), such that when you apply T to the v (T refers to a linear transformation, with domain V and codomain W), T(v)=the zero vector in the codomain W Nullspace of a matrix, is when you let vector space V (domain) be the set of all column matrices, size n*1 (this is the 'home' of the solution x in Ax=0 !!), vector space W (codomain) be the set of matrices, size m*1 v be the column matrix x (the solution to Ax=0), T(v) be the linear transformation function which maps from x-->Ax (the rule of T), where A is a m*n matrix that's left-multiplied onto the x and the zero vector in codomain W is the zero matrix :)
On the independence of the solution vectors, I think it becomes clear when we consider that the free variables (y and t here) always appear on their own in the solution vector ( (2y+6t y -2t t) here). (The pivot variables x and z appear as linear combinations of the free variables in the solution vector). We can write the solution vector as a linear combination of vectors by letting y=0, t=1 and then y=1, t=0 guaranteeing that the vectors are independent.
I'm trying to figure out all the relations of spaces, is the following correct? space time ⊃ outer space ⊃vector space ⊃ space ⊃ (nasa ∪ esa ∪ jaxa) space exploration ⊃ spacex ⊃ sub space ⊃ personal space ⊃ null space ⊃ monopoly board space (iI just got the 'go to jail, go directly to jail, do not pass go, do not collect $200' card, darn it!) Thanks for your videos enjoying both mathematics and silliness!
Dr. Peyam Can you please make a video explaining.. Pointwise and uniform convergence of series , explaining their differences and their relation to continuity with examples.. It would be really helpful.. Because I am having a hard time understanding the idea behind them.
vijayan K my understanding is: uniform convergence is a substype of pointwise convergence. A series of functions is pointwise convergent if any given x can result in a convergent series (in other words, pick a value for x, like five, and see if the series converges; you can pick a different epsilon depending on the x). Uniform convergence says you can't pick any special x when showing the series converges; there has to be one epsilon that fits every choice of x. It's been a few years so I might be off.
Hi Dr. Peyam, can you prove why the construction of the Null space will always be a linearly independent set? My textbook does not give a proof on this one.
I think it becomes clear when we consider that the free variables (y and t here) always appear on their own in the solution vector ( (2y+6t y -2t t) here). (The pivot variables x and z appear as linear combinations of the free variables in the solution vector). We can write the solution vector as a linear combination of vectors by letting y=0, t=1 and then y=1, t=0 guaranteeing that the vectors are independent.
I always thought it was odd that linear algebra uses different terminology that most of algebra. Null Space rather than the Kernel. I suspect it's because linear algebra developed before abstract algebra.
@@drpeyam thank you for your quick reply! Your attention and care that you show towards your followers is why I (we) respect you so much 😁. I (we) cannot wait for your upcoming videos.
I don't get it. The definition is so simple, I thought we'd just solve for (x,y,z,t) vectors. But then strange things arose, an unknown notation with a vertical bar, undefined terms like pivot, modifying rows with a recipe I never heard of ... Pretty sure the result is the same with a naive method, but where did the method you used come from ? Is there a video somewhere I should see ? Please :)
@@drpeyam I got it now, thanks ! I feel stupid, it's what I had been taught very long ago, but with a fancy layout and wording. I didn't recognize the lady at first because of the makeup, but she's the same.
You can set it to whatever value you want and not get a contradiction. Let's say you have an equation y+x=0, you have one free variable. You can set one of those two variables to whatever you want. You don't have two free variables because once you decide what one variable is you can compute the other, you don't have the ability to choose the value of the other variable, the value of the other variable is determined. Now, the reason why he chose y and t to be his free variables is probably because he had 1 for x and 1 for z which means it was easy and convenient for him to express x and z via y and t. He could have chosen x and z to be free variables but it is easier to choose the other 2 variables to be free variables and then express x and z via free variables as x and z have the coefficient 1 in front of them.
The point is, you can express y and t via x and z and have x and z be free variables and get the correct result. The reason why he didn't do it is because it was more convenient to express x and z via t and y and have y and t as free variables because x and z have the coefficient 1 in front of them. I hope that this clears out any confusion.
I really appreciate your teaching style. It is such a refreshing change from most math videos!
Dr. Peyam, your teaching style is awesome. Love the energy!
Thank you!!
You bring a breath of fresh air to a difficult subject! Thank you!❤
Thank you for the video!!
I just learnt about the nullspace, range and rank-nullity theorem in the more general context of linear transformation functions. Before that, we learnt about the rank-nullity theorem in matrices only...I think I finally got the link to the special case where we look only at matrices
Our notes defined nullspace as N(T)=the set of all the vectors v in vector space V (the domain), such that when you apply T to the v (T refers to a linear transformation, with domain V and codomain W), T(v)=the zero vector in the codomain W
Nullspace of a matrix, is when you let
vector space V (domain) be the set of all column matrices, size n*1 (this is the 'home' of the solution x in Ax=0 !!),
vector space W (codomain) be the set of matrices, size m*1
v be the column matrix x (the solution to Ax=0),
T(v) be the linear transformation function which maps from x-->Ax (the rule of T), where A is a m*n matrix that's left-multiplied onto the x
and the zero vector in codomain W is the zero matrix :)
prof there should be a -6 at around 5:43 for the first column 5th component of the column.
Also noticed this.
Thought I was losing my mind lol
EDIT: He fixed it
On the independence of the solution vectors, I think it becomes clear when we consider that the free variables (y and t here) always appear on their own in the solution vector ( (2y+6t y -2t t) here).
(The pivot variables x and z appear as linear combinations of the free variables in the solution vector). We can write the solution vector as a linear combination of vectors by letting y=0, t=1 and then y=1, t=0 guaranteeing that the vectors are independent.
This was a nice video. Educational and puts a smile on my face.
I enjoyed the video, but at 5:43 you might have made an error.
You teach awesome👍👍👍👍
Excellent Teaching Style , Superb Sir :)
I'm trying to figure out all the relations of spaces, is the following correct?
space time ⊃ outer space ⊃vector space ⊃ space ⊃ (nasa ∪ esa ∪ jaxa) space exploration ⊃ spacex ⊃ sub space ⊃ personal space ⊃ null space ⊃ monopoly board space
(iI just got the 'go to jail, go directly to jail, do not pass go, do not collect $200' card, darn it!)
Thanks for your videos enjoying both mathematics and silliness!
Thanks Dr. Peyam
that singing hahahhahahha you will always be my basis~
just amazing thanks a lot sir
You fixed it! Nevermind!
Wonderful thanks alot doctor peyam
Dr. Peyam Can you please make a video explaining.. Pointwise and uniform convergence of series , explaining their differences and their relation to continuity with examples.. It would be really helpful.. Because I am having a hard time understanding the idea behind them.
There’s a video on covering compactness and uniform continuity. Not quite what you want but a good start
vijayan K my understanding is: uniform convergence is a substype of pointwise convergence. A series of functions is pointwise convergent if any given x can result in a convergent series (in other words, pick a value for x, like five, and see if the series converges; you can pick a different epsilon depending on the x). Uniform convergence says you can't pick any special x when showing the series converges; there has to be one epsilon that fits every choice of x. It's been a few years so I might be off.
@@drpeyam Dr peyam can you suggest a good read..
www.personal.psu.edu/auw4/M401-notes1.pdf
math.byu.edu/~bakker/M341/Lectures/Lec28.pdf
bro is cracked at matrices.
Hi Dr. Peyam, can you prove why the construction of the Null space will always be a linearly independent set? My textbook does not give a proof on this one.
I think it becomes clear when we consider that the free variables (y and t here) always appear on their own in the solution vector ( (2y+6t y -2t t) here).
(The pivot variables x and z appear as linear combinations of the free variables in the solution vector). We can write the solution vector as a linear combination of vectors by letting y=0, t=1 and then y=1, t=0 guaranteeing that the vectors are independent.
When you started the geometric interpretation i thought somebody was going to draw time!!!
In Holland we also say NUL for 0, thnx for your lecture
I always thought it was odd that linear algebra uses different terminology that most of algebra. Null Space rather than the Kernel. I suspect it's because linear algebra developed before abstract algebra.
I think it’s to distinguish it from linear transformations, although they’re the same
in german it is also "Kern" in linear algebra and not "Nullraum" or something
How do you know by looking at the matrix in RREF that y and t are going to be free variables? is it their lack of a pivot?
Yes basically; the free variables are in the non-pivot columns
@@drpeyam thank you for your quick reply! Your attention and care that you show towards your followers is why I (we) respect you so much 😁. I (we) cannot wait for your upcoming videos.
How is this comment from a week ago but the video was just uploaded?
It’s been unlisted for a week (so technically available), but officially released today
I don't get it. The definition is so simple, I thought we'd just solve for (x,y,z,t) vectors. But then strange things arose, an unknown notation with a vertical bar, undefined terms like pivot, modifying rows with a recipe I never heard of ... Pretty sure the result is the same with a naive method, but where did the method you used come from ? Is there a video somewhere I should see ? Please :)
Yeah! Check out my Systems of Equations (Lay Chapter 1) playlist. It’s called Gaussian Elimination
@@drpeyam I got it now, thanks ! I feel stupid, it's what I had been taught very long ago, but with a fancy layout and wording. I didn't recognize the lady at first because of the makeup, but she's the same.
The great one!
What does it mean to be a variable to be free?
You can set it to whatever value you want and not get a contradiction. Let's say you have an equation y+x=0, you have one free variable. You can set one of those two variables to whatever you want. You don't have two free variables because once you decide what one variable is you can compute the other, you don't have the ability to choose the value of the other variable, the value of the other variable is determined.
Now, the reason why he chose y and t to be his free variables is probably because he had 1 for x and 1 for z which means it was easy and convenient for him to express x and z via y and t. He could have chosen x and z to be free variables but it is easier to choose the other 2 variables to be free variables and then express x and z via free variables as x and z have the coefficient 1 in front of them.
The point is, you can express y and t via x and z and have x and z be free variables and get the correct result. The reason why he didn't do it is because it was more convenient to express x and z via t and y and have y and t as free variables because x and z have the coefficient 1 in front of them. I hope that this clears out any confusion.
At 5:20 , R1-3R2 ->R1 should be [-1, -2, 0, -6] not [-1, -2, 0, 0]
That’s mentioned in the comments already
@@drpeyam Sorry, I could not find it. Anyway, congratulations for the great explanation!
thank you
The first 2 columns is multiplied by -2
The Ker(T) is a null space?
Yes same thing
And in english null and VoId menans without Value
You will always be my Baby😂😂😂😂😂😂
@5:43 I think you made a mistake
ma man
in polish we say "JĄDRO" which also means ... testicle.😂😂😂😂😂😂😂
"Find the testicle of A"
In Hindi it's "Lul" means zero. So same it is 😀😀😀😀
Ohkkiii😂
Haha in french we use the word Ker(A) and not Nul :D
We use Ker for linear transformations
@@drpeyam ain't it what we are currently studying?
A is a matrix. T(x) = Ax is its linear transformation.
Yes, but it’s just to distinguish kernels for matrices from kernels of general linear transformations (on infinite dimensional vector spaces)
In my study, with the above definition I have, we would have ker(T) = null(A).
Nul ! Nul ! Nul ! Germain !