I have used the Kirchhoff's Law for 6 yrs since studied electrical engineering, and that's the first time I learn it from perspective of matrices, what a great lecture!
I'm fairly sure the exact same principles applied by Gil to KCL are universal, I could use the same exact approach for building statics if I'm not mistaken.
00:14 Real linear algebra uses matrices from applications with a definite structure. 03:28 Understanding the incidence matrix and its applications 09:39 Real graphs lead to sparse matrices with structure. 12:46 Understanding potential differences in networks 18:35 Understanding the rank and null space of matrices 21:24 Understanding the null space dimension 27:11 Kirk's current law represents a balance equation and conservation law. 30:14 Finding the null space of a matrix using graphs and elimination 35:39 The dimensions of row and column spaces of a transpose matrix 38:29 A graph without a loop is called a tree. 44:26 Understanding the basic equations of electrical circuits and the addition of external sources. 47:07 Matrix a transpose a is always symmetric very good explanation sir
22:14 , *""usually sometimes during the semester i drop one of these erasers behind there. That's a great moment. There's no recovery. There's -- centuries of erasers back there."* I like prof gilbert's sense of humor.
x = x1, x2, x3, x4 are like potentials V1, V2, V3, V4 at each node Ax = 0 gives potential differences (voltages) V2-V1 etc. I = V/R (Ohm's law) gives currents I1 to I5 (y1 to y5) at each node (he uses C to denote R^(-1) matrix) (A^T)y = 0, which is Kirchhoff's current law
I was kinda lost through this video, wondering why he was calculating dimensions of the four subspaces of this graph matrix, while also guessing why he chose the edge matrix A the way he did(As a CS student I'm used to seeing graph in a different way). Then literally at the final minute he explained how the transformation worked and I was overjoyed for having an epiphany. I love linear algebra now.
my math class is using the textbook from Strang atm and these videos pretty much cover the concepts from it. I'm so glad I found this because I can't understand my math teacher at all because of his thick accent. I'm paying for an education but I'm getting it from here.
Lectures like this one make me regret not taking Linear Algebra in college. However, I would probably never have come across Prof Strang's lectures had I did. Such a great teacher for showing how interesting and useful a subject that I once thought was so dull.
Knowing where I ended up, I would have gotten some boring, overworked grad student who didn't know how to lecture. I guess waiting all these years before studying this paid off......
at 46:18 the equation written on the board A(T)CAx = f is missing the minus sign. When we calculate the potential drop e, AX is negative because potential is dropping, and if there is a Voltage Source b connected then e = b - AX. Now Balance equation becomes A(T)CAX = -f.
This lecture is such a sad story for those who never understood electrical engineering basics like me. It could have gotten a lot better if he stuck to those friendships and Internet applications he mentioned in the last lecture.
watch until the end! Tree graphs are how internet websites and friendships are plotted... usually. So if you know the number of nodes (friends / sites) and edges (relationships) you can find the circles (groups of friends / mutually relatable sites) as it will resolve with the outcome of 1.
DR. Strang thank you for linking linear algebra to real world technology. Circuit design and machine design uses linear algebra. The development of Google uses linear algebra.
I high-key thought it was hella cool when he added the two loops together on the picture, subsequently adding the two basis vectors for N(A)^T and creating a dependent vector. I was kind of expecting that to happen just theoretically, but to actually see it happen just blew my puny mind.
I thought this was an excellent lecture - lectures10 and 11 were a real struggle to get through but this really cleared up a lot of the terminology and why the four fundamental subspaces are so important. Thumbs up Professor Strang. (Weird ironing creases on his blue shirt in the vicinity of his right nipple.)
7:40 Nodes 1,2, and 3 form a loop? It seems strange since the direction of the edge connecting nodes 1 and 3. At least from my usual use of the word "loop" it doesn't seem like a loop since if you leave node 1 you can't get back to node 1. Comments anyone?
+Adam Levin I can see how this may be confusing, especially if you aren't very practiced with evaluating DC circuits. The directions of the arrows are completely arbitrary. The professor could have just as easily switched the direction of some or all of the arrows (it would have changed the signs of some of the terms in the matrix from + to - or vice versa). The arrows aren't meant to indicate a direction of flow; they're just meant to provide a +/- convention for you to write your matrix. Flow could be in the direction opposite the arrow, it would be negative flow, and when using the +/- convention to write your matrix, the math would calculate it to be such. I hope this answers your question. Let me know if I just made things worse.
If the direction of the arrows have nothing to do with the flow, couldn't I randomly assign them +- signs? If yes, wouldn't that result in a totally different A-matrix for the same graph? Doesn't that throw the whole lecture's point off? What am I missing?
I think you are thinking from the perspective of probabilistic graphical models where dependence flows through the arrows which is not the case here since the arrows just show the positive direction for flow
Am I the only one who finds this lecture somewhat confusing? I mean I get the fundamentals, like how column and row ranks are equal (denoted r), and how the ranks (or dimensions) of the null spaces of A and A^T are (n-r) and (m-r) respectively. But that's pretty much all I got. I didn't quite understand how he jumped to Euler's formula, especially when he stated that the number of loops is always equal to the dimensions of N(A^T).
@tbarker5252 So you are saying is that the graph with no loops he drew at about 38:00 is not a tree? He didn't give any definition, he said "A tree is the name for a graph with no loops" meaning that a graph with no loops is a tree, which is absolutely correct, no matter how you look at it! You certainly don't need any degrees to see that ;)
great lecture as always, I got lost when it comes Euler Equation. don't quite understand how column, rows become nodes, edges, and why do we need #node -1 in the equation. The answer is probably in the matrices representation of the graph. 248k views for this lecture, last one was close to 300k. lets see how many views is next lecture.
if you compare it to the formula you mentioned above, the number of faces in the planar graph would not be 2 but 3 because the outer region is also considered to be a face of the graph, i guess you could imagine the graph being drawn on a sphere. the region outside what you might call the "figure" would be considered to be another face. so i think his definition of the number of loops is consistent with the formula.
This is because he only took A=[1,1,1,1] to be a nullspace, but there is one as well which is [0,0,0,0] a zero vector. You can see that pattern when he counted the loops. He only took the ones which were inside the graph boundary, but the whole graph boundary is a loop as well, so the total sum of loops are 4 not 3 which will get that 2 in Euler's formula.
Math is not my major, but it seems if we count the outer (infinitely large) region in a planer graph as a face, we get the Euler number of 2. In the example of this lecture, the outer region is not counted, so the Euler number would be 1. Except for the outer face, each loop (i.e., each vector of the basis for N(A Transpose)) makes one face, so the number of loops and the number of inner faces of a planer graph should be the same.
Hi: I think in the example talks about electric Networks, so LOOP is correct. In fact OPEN LOOP and CLOSED LOOP is used frecuently in electric Networks or in Control Systems to study things that Stability,etc. Cycle is used in the time domain to know the frecuency and in fact, the frecuency is measured in CYCLES/SEG or Hz.
Is there is a convention for whether an incidence matrix for a graph is edges x nodes or nodes x edges? In this video its edges X nodes but alot of other videos I see nodes X edges Thanks for your help.
Solving Ax = b for a graph you always get x1 = x2 = ... = xn and then you have only one special solution: for instance, vector [1 1 1 1]. So, the dimension of nullspace is 1, you have n unknowns, then the rank = n - 1.
Since the dimension of left null space is the number of loops, the rank is the number of edges that do not form a loop, and that number is always 1 less than the number of nodes
Let me try to answer this physically: we know that rank = n - dim N(A). How do we know dim N(A)? That's the dimension of the solution to Ax=0, physically that means we're looking for the solution to when there's no current in the circuit. There's only one (bunch of ) solution to this scenario: when the potential of every node are equal, thus the solution of Ax=0 is C[1111] transpose, that's one dimensional, so dim N(A)=1
My intuition is that the rank in this graph application equals the number of edges required in a tree (which has no loop). Every tree with n nodes always has n-1 edges.
So nodes are like some 0 dimensional thing; edges are 1 dimensional since they connect nodes; loops are 2 dimensional since they have area. I never understood that weird Eular's formula, finally linear algebra shed some lights on it ~
Great lecture! However I have a doubt, can someone please tell me how to find the basis from row reduced echelon form? The row reduced matrix according to me R = [(1 0 1 0 0), (0 1 1 0 -1), (0 0 0 1 1), (0 0 0 0 0)] I have gone through lectures 1-11, but still I am not getting it. Am I missing something? Please help
Basis vectors in case of null space would be the vectors which satisfy ax=0 by putting free variables in equation to 0 and 1 respectively and solving for rest of the variables
@Abmaj7add2 In this perspective, look at the matrix by rows, every row corresponds to an edge. And the first three rows, if see it in the graph, make a loop.
Electric current flows from (+) to (-), from higher potential to lower potential. But starting node of each edge is marked with -1 in the matrix and end node marked with +1. Why not other way around?
Skins Song Electric current as a concept is thought as flowing from + to - , while actual charged particles are flowing according to their charge. So electrons (as negatively charged) move to + but positively charged ions move to -
Absolutely! Now I remember, thanks for the explanation. Guess I was shocked by the mathematical version of simple physical phenomenon. Anyway, sorry that I can't answer your question.
I am kind of confused about the y=Ce=CAx in this lecture. How we can draw this conclusion from the process of resolving the equation instead of relying on the physical explanation?
They form a loop in the sense that the edges make a closed shape. The arrows don't indicate flow direction, they are just arbitrary so we know where to put the pluses and minuses in the matrix. It's entirely possible for current to flow against the direction of the arrow, we just say the current is negative in that case.
Also, as he said, the external force term, f, adds non-equilibrium, and time dependence. In the parlance of differential equations, a non homogeneous system of differential equations
I guess he means independent loops. So in the 4 nodes, 5 edges example given in the lecture, you, may very well consider the larger loop(1,2,3,4), but then the second loop would have to be either 1,2,3 or 1,3,4 since counting both makes one of the three loops dependen.t
My god😂😂 just imagine how messed up it would be to teach it in matrix form and them let students discover that current in a closed loop follows the law and say that that's how the law came about. 😂😂😂 This is why I love math math has its own causality.
For a matrix A, its columns represent nodes while rows represent edges. We know that current is specified for an edge and potential is specified for a node. By finding out null-space of A transpose, you are basically taking linear combinations of columns which equals zero. And the columns of A transpose are edges which means we are finding out relation between edges. And current is represented by edge, so I guess its all about relating nodes and edges to potential and current.
@@hyungjoonpark83 Ah not really anymore, at the time, I didn't know that he's demonstrating Kirchoff's law as an example, I was like what is Kirchoff's law doing here. (I didn't know what Kirchoff's law is at the time and as for someone who is not native speaker that didn't have any physic lesson in English before, it's quite confusing.)
We know that the first three rows of A form a loop. So, the rank of the matrix should be 2, right? We know that there are maximum 2 rows that are independent in A. Please explain.
the first three rows form a loop, that means there are only 2 independent rows in there. But there are another rows, and from there, there is one other independent row. It makes the matrix rank 3. What he tries to emphasize is the first rows/columns are not always the pivots one (the independent ones together)
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I have used the Kirchhoff's Law for 6 yrs since studied electrical engineering, and that's the first time I learn it from perspective of matrices, what a great lecture!
Yeah this lecture revealed a whole new dimension of understanding the KCL!
sure -
I'm fairly sure the exact same principles applied by Gil to KCL are universal, I could use the same exact approach for building statics if I'm not mistaken.
How tho, I thought they are teaching elimination with matrices for kirchoff's law in electrical circuits courses
the most beautiful and natural view of KCL I've ever seen😍
00:14 Real linear algebra uses matrices from applications with a definite structure.
03:28 Understanding the incidence matrix and its applications
09:39 Real graphs lead to sparse matrices with structure.
12:46 Understanding potential differences in networks
18:35 Understanding the rank and null space of matrices
21:24 Understanding the null space dimension
27:11 Kirk's current law represents a balance equation and conservation law.
30:14 Finding the null space of a matrix using graphs and elimination
35:39 The dimensions of row and column spaces of a transpose matrix
38:29 A graph without a loop is called a tree.
44:26 Understanding the basic equations of electrical circuits and the addition of external sources.
47:07 Matrix a transpose a is always symmetric
very good explanation sir
He completely changed my view on linear algebra. The whole thing is just amazing!
I did a maths degree, I've know about Kirchoff's Rule, and I studied linear algebra. No one explained how it was all connected like this.
that is why this course is taught in MIT, and MIT is the best engineering school worldwide.
Same. This lecture is the beginning of it all being brought together.
@@boyuanxie2708 Don't forget Caltech (JPL) and Stanford (Silicon Valley). Holy Trinity!
22:14 , *""usually sometimes during the semester i drop one of these erasers behind there. That's a great moment. There's no recovery. There's -- centuries of erasers back there."* I like prof gilbert's sense of humor.
right, the way that's worded is precisely my kind of humor
You must be fun at parties
LOL, let me show you guys what is called uninvertible, see, it doesn't matter you do a transpose or anything else, it's uninvertible.
@@kreglfromworld you must be fun if and only if at a party
Imagine 100 years later, Prof Strang's lost erasers might become some sort of MIT historic items. Hahaha
x = x1, x2, x3, x4 are like potentials V1, V2, V3, V4 at each node
Ax = 0 gives potential differences (voltages) V2-V1 etc.
I = V/R (Ohm's law) gives currents I1 to I5 (y1 to y5) at each node
(he uses C to denote R^(-1) matrix)
(A^T)y = 0, which is Kirchhoff's current law
"Euler again ! that guy never stopped"
This sentence just killed me xD
dude, you nailed it. :P
Euler... man, he was such a nerd. XD
What will you tell about newton?😂
we cant do nothing about him
Such nice illustrations of applications of the theoretical concepts presented thus far. One more teaching masterclass from prof. Strang!
4:37
Kirchoff's Laws || Potential, Potential Difference, Currents || An electrical network
5:10
Hydraulic network || Flow of water, Flow of oil || Design for a structure || Bridge || Dome
5:15
INCIDENCE Matrix
5 Rows are 5 Edges
I was kinda lost through this video, wondering why he was calculating dimensions of the four subspaces of this graph matrix, while also guessing why he chose the edge matrix A the way he did(As a CS student I'm used to seeing graph in a different way). Then literally at the final minute he explained how the transformation worked and I was overjoyed for having an epiphany. I love linear algebra now.
A brand new view of the world
As an electronic engineer I absolutely loved this class, just amazing!
This lecture was beautiful... The way everything connects and binds together is stunning... Invaluable knowledge! Thanks mr. Strang
prof gilbert strang : teaching the beauty of linear algebra
me : wondering how we are going to get those erasers stuck behind the backboard ........
Best Class I have ever seen...Wish I could be one of the students sitting in the classroom.
my math class is using the textbook from Strang atm and these videos pretty much cover the concepts from it. I'm so glad I found this because I can't understand my math teacher at all because of his thick accent. I'm paying for an education but I'm getting it from here.
I don't usually write comments but this is one of, if not the best, lecture if ever been a part of. Thanks MIT!!
Great lecture from one of the grandmasters...indeed a true honor to hear this
This lecture is what MIT means in this world!
Never knew all these things are connected so beautifully ! Thanks prof. 🙏
The way he says the days of the week is perfect. "Fridee"
Lectures like this one make me regret not taking Linear Algebra in college. However, I would probably never have come across Prof Strang's lectures had I did. Such a great teacher for showing how interesting and useful a subject that I once thought was so dull.
Knowing where I ended up, I would have gotten some boring, overworked grad student who didn't know how to lecture. I guess waiting all these years before studying this paid off......
I am overjoyed by this lecture!! A beautiful system of linear (in)dependencies.
This is like a trip down memory lane for anyone that studied Electrical Engineering. Freshman year, when things were simple 😂
Wow this really opened the doors to mixing differential equations and matrices in a way differential equations classes dont teach.
You have to admit that this course makes you realise youtube comment sections can be good too!
Didn't notice that until you mentioned. No toxic comments at all. I don't think I ever seen this on a yt comment section.
This lecture is just like a philosophical adventure of applied linear algebra.
at 46:18 the equation written on the board A(T)CAx = f is missing the minus sign. When we calculate the potential drop e, AX is negative because potential is dropping, and if there is a Voltage Source b connected then e = b - AX. Now Balance equation becomes A(T)CAX = -f.
how did you know that AX would be negative?
Five Minutes of the "GRAPHS" is in lecture 11. Just follow lecture 11 from 39:30.
linear algebra should be one of the first course taught in freshman together with calculus before physics and circuit analysis
Beautiful connection to the cycle space of a graph. Thank you!
Hellow
Where are you from
Great lecture gilbert strang..
You are my saviour for final exams.
Respect
are you still studying?
This lecture is such a sad story for those who never understood electrical engineering basics like me. It could have gotten a lot better if he stuck to those friendships and Internet applications he mentioned in the last lecture.
watch until the end! Tree graphs are how internet websites and friendships are plotted... usually. So if you know the number of nodes (friends / sites) and edges (relationships) you can find the circles (groups of friends / mutually relatable sites) as it will resolve with the outcome of 1.
DR. Strang thank you for linking linear algebra to real world technology. Circuit design and machine design uses linear algebra. The development of Google uses linear algebra.
This must be the best lecture I have ever had.
20:00 a rationale for determining rank (using null space) so that you don't end up with a junk matrix in an engineering application.
I high-key thought it was hella cool when he added the two loops together on the picture, subsequently adding the two basis vectors for N(A)^T and creating a dependent vector.
I was kind of expecting that to happen just theoretically, but to actually see it happen just blew my puny mind.
I thought this was an excellent lecture - lectures10 and 11 were a real struggle to get through but this really cleared up a lot of the terminology and why the four fundamental subspaces are so important. Thumbs up Professor Strang.
(Weird ironing creases on his blue shirt in the vicinity of his right nipple.)
This is the greatest lecture of math i have ever attended
My mind was blown two times in just 45 minutes ❗❗🤯🤯
This class was amazing, just amazing.
Genius!!!!!!!!!!!!!!!! mind-blowing lesson! thank you professor!!!!!!!!!!!!!!!!!!!!!!!!!
what a great man
7:40 Nodes 1,2, and 3 form a loop? It seems strange since the direction of the edge connecting nodes 1 and 3. At least from my usual use of the word "loop" it doesn't seem like a loop since if you leave node 1 you can't get back to node 1. Comments anyone?
+Adam Levin Confusing also because in graph theory a loop seems to be an edge that connects a node to itself..
+Adam Levin I can see how this may be confusing, especially if you
aren't very practiced with evaluating DC circuits. The directions of the
arrows are completely arbitrary. The professor could have just as
easily switched the direction of some or all of the arrows (it would
have changed the signs of some of the terms in the matrix from + to - or
vice versa).
The arrows aren't meant to indicate a direction of flow; they're just meant to provide a +/- convention for you to write your matrix. Flow could be in the direction opposite the arrow, it would be negative flow, and when using the +/- convention to write your matrix, the math would calculate it to be such.
I hope this answers your question. Let me know if I just made things worse.
i read the comments cause i had the same question, that makes perfect sense, thank you. now i get it
If the direction of the arrows have nothing to do with the flow, couldn't I randomly assign them +- signs? If yes, wouldn't that result in a totally different A-matrix for the same graph? Doesn't that throw the whole lecture's point off? What am I missing?
I think you are thinking from the perspective of probabilistic graphical models where dependence flows through the arrows which is not the case here since the arrows just show the positive direction for flow
Graph and Tree are concepts used in Computer Science frequently
42:27 we see Euler again, that guy never stops 😂✨
Nodes - Edges + Loops = 1 must be assuming a connected graph since you can just have 3 unconnected Nodes and its obviously not true ;)
28:47 "PHYSICISTS BE OVERJOYED RN!!"
This blew my mind. It was Awesome!
Am I the only one who finds this lecture somewhat confusing? I mean I get the fundamentals, like how column and row ranks are equal (denoted r), and how the ranks (or dimensions) of the null spaces of A and A^T are (n-r) and (m-r) respectively. But that's pretty much all I got.
I didn't quite understand how he jumped to Euler's formula, especially when he stated that the number of loops is always equal to the dimensions of N(A^T).
this was the best lecture in my life .. so far ))
Linear algebra is wild
Amazing Lecture Professor, well done!
Prof Stang The GOAT.
@tbarker5252 So you are saying is that the graph with no loops he drew at about 38:00 is not a tree? He didn't give any definition, he said "A tree is the name for a graph with no loops" meaning that a graph with no loops is a tree, which is absolutely correct, no matter how you look at it! You certainly don't need any degrees to see that ;)
great lecture as always, I got lost when it comes Euler Equation. don't quite understand how column, rows become nodes, edges, and why do we need #node -1 in the equation. The answer is probably in the matrices representation of the graph. 248k views for this lecture, last one was close to 300k. lets see how many views is next lecture.
# node - 1 is just the rank, since there are n columns (1 for each node), and the nullspace is one-dimensional. Thus, r = n-1 = # nodes - 1
I'm confused. I think the euler's formula was V - E +F = 2 not 1.
if you compare it to the formula you mentioned above, the number of faces in the planar graph would not be 2 but 3 because the outer region is also considered to be a face of the graph, i guess you could imagine the graph being drawn on a sphere. the region outside what you might call the "figure" would be considered to be another face. so i think his definition of the number of loops is consistent with the formula.
This is because he only took A=[1,1,1,1] to be a nullspace, but there is one as well which is [0,0,0,0] a zero vector. You can see that pattern when he counted the loops. He only took the ones which were inside the graph boundary, but the whole graph boundary is a loop as well, so the total sum of loops are 4 not 3 which will get that 2 in Euler's formula.
@@mirzaxanaliyev7211 Thank you nigga
Math is not my major, but it seems if we count the outer (infinitely large) region in a planer graph as a face, we get the Euler number of 2. In the example of this lecture, the outer region is not counted, so the Euler number would be 1. Except for the outer face, each loop (i.e., each vector of the basis for N(A Transpose)) makes one face, so the number of loops and the number of inner faces of a planer graph should be the same.
Can someone please explain the significance of the answer to his last question at 47:13 ?
Hi:
I think in the example talks about electric Networks, so LOOP is correct. In fact OPEN LOOP and CLOSED LOOP is used frecuently in electric Networks or in Control Systems to study things that Stability,etc.
Cycle is used in the time domain to know the frecuency and in fact, the frecuency is measured in CYCLES/SEG or Hz.
I have to say, this one is pretty challenging for me, cuz I have no idea about Kirchoff's Law.
yk it's 4am and basis,subspaces have messed me up, i didn't get the last 2 lecs completely and maybe it is my fault but on this one i bouced back
Subgraph can be a loop or cant be without a loop. Did this answer ur question?
Is there is a convention for whether an incidence matrix for a graph is edges x nodes or nodes x edges?
In this video its edges X nodes but alot of other videos I see nodes X edges
Thanks for your help.
Any intuition for the rank to be n-1 ??
Solving Ax = b for a graph you always get x1 = x2 = ... = xn and then you have only one special solution: for instance, vector [1 1 1 1]. So, the dimension of nullspace is 1, you have n unknowns, then the rank = n - 1.
Since the dimension of left null space is the number of loops, the rank is the number of edges that do not form a loop, and that number is always 1 less than the number of nodes
Let me try to answer this physically: we know that rank = n - dim N(A). How do we know dim N(A)? That's the dimension of the solution to Ax=0, physically that means we're looking for the solution to when there's no current in the circuit. There's only one (bunch of ) solution to this scenario: when the potential of every node are equal, thus the solution of Ax=0 is C[1111] transpose, that's one dimensional, so dim N(A)=1
My intuition is that the rank in this graph application equals the number of edges required in a tree (which has no loop). Every tree with n nodes always has n-1 edges.
So nodes are like some 0 dimensional thing;
edges are 1 dimensional since they connect nodes;
loops are 2 dimensional since they have area.
I never understood that weird Eular's formula, finally linear algebra shed some lights on it ~
Thanks Xintong
This lecture made my brain hurt, probably because I don't know anything about physics.
We're in the same boat even if we're 5 years apart!
We're in the same boat even if we're 5 and 3 weeks years apart!
@@hyungjoonpark83 Here I am, 7 years apart xD
Same boat from 2020
11.07.2020 - me understanding you from 2013. Have you learned anything about phisics since this comment?
Great lecture! However I have a doubt, can someone please tell me how to find the basis from row reduced echelon form?
The row reduced matrix according to me R = [(1 0 1 0 0), (0 1 1 0 -1), (0 0 0 1 1), (0 0 0 0 0)]
I have gone through lectures 1-11, but still I am not getting it. Am I missing something? Please help
Basis vectors in case of null space would be the vectors which satisfy ax=0 by putting free variables in equation to 0 and 1 respectively and solving for rest of the variables
@Abmaj7add2 In this perspective, look at the matrix by rows, every row corresponds to an edge. And the first three rows, if see it in the graph, make a loop.
42:21 "Euler never stop" lol
Electric current flows from (+) to (-), from higher potential to lower potential. But starting node of each edge is marked with -1 in the matrix and end node marked with +1. Why not other way around?
Mark me if I'm wrong, I learned physics more than ten years ago, but I remembered that electrons move from (-) to the (+)
Skins Song Electric current as a concept is thought as flowing from + to - , while actual charged particles are flowing according to their charge. So electrons (as negatively charged) move to + but positively charged ions move to -
Absolutely! Now I remember, thanks for the explanation. Guess I was shocked by the mathematical version of simple physical phenomenon. Anyway, sorry that I can't answer your question.
He implied that while graphs are discrete, there is something analogous to a graph which is continuous. What is that something?
Alex Dowad
Normal graphs in algebra
Why is the sum of the elements of a vector in the null space equal to zero?
Was launching matlab to play fifteen, prof immediately speaks of matlab. How did you know 11 years prior!?
I'm new to math but isn't Euler's formula e^ix = cosx + isinx,
I can't see how the equations with nodes and edges relate to the formula..
For now I'm guessing they're two different formulas..
Euler was a busy man....
Euler had many formulae credited to his name
23:20
Hi tbarker5252,
Do you think that the word cycle and loop are different?
And this discussion contribute with some development ... I dont think so.
THIS MAN IS A FUCKING GENIUS
dont mind me but why is A^TA symmetriiiiiiic
symetric means a equals a^t lets take traspose of a^ta
(a^ta)^t = a^ta^tt and a^tt = a
and you get a^ta = (a^ta)^t
did you understand 12th lesson ?
I am kind of confused about the y=Ce=CAx in this lecture. How we can draw this conclusion from the process of resolving the equation instead of relying on the physical explanation?
At the end of this lecture, come to the equation ATCAx=f. You may think this as the introduction of later lectures about eigenvalue decomposition.
Great lecture!
Wow! Euler characteristics just came out of nowhere.
7:45 how is edge 1, 2 and 3 make a loop when loop 3 is pointing to the opposite direction to the flow of edge 1 and 2 ??
They form a loop in the sense that the edges make a closed shape. The arrows don't indicate flow direction, they are just arbitrary so we know where to put the pluses and minuses in the matrix. It's entirely possible for current to flow against the direction of the arrow, we just say the current is negative in that case.
Does the last equation (At*y=f) add any new information? I suppose find y (currents) from C*e would be last step right?
Yes the y vector describes how the current is distributed in the system and f denoted the amount of external current applied.
Also, as he said, the external force term, f, adds non-equilibrium, and time dependence. In the parlance of differential equations, a non homogeneous system of differential equations
What if Mr Strang has to take square root of r
I Really Like The Video Graphs, Networks, Incidence Matrices From Your
Beautiful... lovely..
how can I like a video twice?
is it possible that i didn't get anything because i do not know physics
Thanks sir
At 43:22 why did we not consider the bigger loops?
We consider only smaller loops(or faces) in Euler's formula which applies to planar graphs.
I guess he means independent loops. So in the 4 nodes, 5 edges example given in the lecture, you, may very well consider the larger loop(1,2,3,4), but then the second loop would have to be either 1,2,3 or 1,3,4 since counting both makes one of the three loops dependen.t
My god😂😂 just imagine how messed up it would be to teach it in matrix form and them let students discover that current in a closed loop follows the law and say that that's how the law came about. 😂😂😂
This is why I love math math has its own causality.
Why are we doing A transpose to fine the current? Any intuition?
For a matrix A, its columns represent nodes while rows represent edges. We know that current is specified for an edge and potential is specified for a node. By finding out null-space of A transpose, you are basically taking linear combinations of columns which equals zero. And the columns of A transpose are edges which means we are finding out relation between edges. And current is represented by edge, so I guess its all about relating nodes and edges to potential and current.
@@parikhrachit4061 Yeah got it. Thanks a lot , appreciate it man.
@@parikhrachit4061 thanks
This 12th lecture is really confusing me.
@@hyungjoonpark83 Ah not really anymore, at the time, I didn't know that he's demonstrating Kirchoff's law as an example, I was like what is Kirchoff's law doing here. (I didn't know what Kirchoff's law is at the time and as for someone who is not native speaker that didn't have any physic lesson in English before, it's quite confusing.)
We know that the first three rows of A form a loop. So, the rank of the matrix should be 2, right? We know that there are maximum 2 rows that are independent in A. Please explain.
the first three rows form a loop, that means there are only 2 independent rows in there. But there are another rows, and from there, there is one other independent row. It makes the matrix rank 3. What he tries to emphasize is the first rows/columns are not always the pivots one (the independent ones together)
Pure math man
euler is genius
If only I had seen this video before my third uni maths assignment T_T
thank a lot
I wish I did watched it at 2000.
how did he get the numbers for the matrix at 9:35?