Dear linear algebra students, This is what matrices (and matrix manipulation) really look like

Summary

This video, sponsored by Brilliant, explores the fascinating world of matrices and their applications beyond traditional math curriculums. The presenter delves into matrix manipulation using 3D software, illustrating how matrices can represent systems of equations and be visualized as intersecting planes or vectors. The video explains concepts like null space, row space, and column space in a 3D context, and connects these mathematical ideas to real-world applications such as circuit analysis. It also touches on graph theory, demonstrating how matrices can represent and simplify complex networks. The video encourages viewers to think visually about linear algebra and to explore Brilliant's courses for a deeper understanding of these concepts and their applications.

Takeaways

• 😀 The video is sponsored by Brilliant, an educational platform that offers courses on various subjects including linear algebra.
• 📊 Matrices can be visualized as sets of vectors, which is useful for understanding systems of equations and their solutions.
• 🔍 Two main ways to approach matrix problems are by finding the intersection of planes or by determining scale factors for vectors to sum to a specific vector.
• 🎯 The null space of a matrix represents all the solutions where the equations equal zero, which can be visualized as a line or a plane depending on the matrix.
• 🔄 Matrix manipulation, such as Gaussian elimination, can simplify solving systems of equations and reveal the underlying structure of the problem.
• 📐 The row space of a matrix is the set of all vectors perpendicular to the null space, and it's always two dimensions less than the matrix's dimension.
• 🔗 The concept of linear dependence and independence of vectors is crucial for understanding the solutions that a matrix can represent.
• 🔍 The incidence matrix is used to represent networks or circuits, and its null space can reveal important properties about the network, like the absence of loops.
• 🌐 The row and column spaces of a matrix are always the same dimension, which is a fundamental property of linear algebra.
• 🔧 Applications of matrices extend beyond traditional mathematics to include graph theory, network analysis, and even the Google page rank algorithm.

Q & A

What is the primary focus of the video sponsored by Brilliant?

The video focuses on exploring matrix manipulation and arithmetic in a visually engaging way, using 3D software and applications not typically taught in schools.

How does the video presenter suggest visualizing a matrix?

The presenter suggests visualizing a matrix as a set of vectors, either as three column vectors or three row vectors, and demonstrates this with 3x3 matrices.

What is the significance of representing a matrix as column vectors in the context of systems of equations?

Representing a matrix as column vectors allows for the system of equations to be visualized as a sum or linear combination of these vectors, where the variables x, y, and z are scale factors.

How does the video explain the solution to a system of equations using matrix visualization?

The video explains that the solution to a system of equations can be visualized as the intersection of planes or as scale factors that add vectors tip-to-tail to reach a specific vector b.

What is the null space in the context of the video?

The null space is the set of all solutions where all the equations equal zero, often represented as an intersection of planes when they are all set to zero.

How does the video demonstrate the use of Gaussian elimination in the context of matrices?

The video shows Gaussian elimination by manipulating the constants in the equations to cancel out variables, which results in a simpler form that is easier to analyze.

What is the relationship between the null space and the row space as explained in the video?

The video explains that the null space and the row space are always perpendicular to each other, and that they have dimensions that add up to the dimension of the matrix.

How does the video connect matrix concepts to graph theory and networks?

The video connects matrix concepts to graph theory by using an incidence matrix to represent a directed graph, and then analyzing the null space and row space in terms of the graph's properties.

What does the video suggest about the physical meaning of the null space in a circuit represented by an incidence matrix?

The video suggests that the null space in a circuit represented by an incidence matrix corresponds to the voltages that result in no current flow, which aligns with Kirchhoff's voltage law.

What additional topics does the video mention that are covered by Brilliant's courses?

The video mentions that Brilliant's courses cover topics such as adjacency matrices, the use of matrices in graph theory, and unique applications like the Google page rank algorithm, as well as differential equations and their applications.

Outlines

00:00

📊 Matrix Manipulation and Visualization

The paragraph introduces the concept of matrices and their manipulations, emphasizing how they can be visualized and understood through 3D software. It discusses the initial struggle with matrices in a school setting and aims to present them in an engaging way. The paragraph explains how matrices can be seen as sets of vectors, specifically focusing on 3x3 matrices, which can be interpreted as either three column vectors or three row vectors. It further elaborates on how these matrices can represent systems of linear equations and how they can be visualized as intersections of planes or as linear combinations of column vectors. The concept of null space is introduced as the set of solutions where all equations equal zero, which can be visualized as the intersection of planes when they are not all zero.

05:01

🔍 Exploring Row and Column Spaces

This section delves into the row and column spaces of matrices, explaining how they are related to the null space and how they can be visualized geometrically. It describes the row space as the set of all vectors perpendicular to the null space, which is represented by the planes formed by the row vectors. The column space, on the other hand, is the set of all possible linear combinations of the column vectors. The paragraph also touches on the concept of linear dependence and how it affects the ability to span the entire space, contrasting this with linear independence. The discussion includes the implications of these spaces for solving systems of equations and how they relate to the physical world, such as in circuits.

10:02

🔌 Applications of Matrices in Circuits

The paragraph explores the application of matrices in the context of electrical circuits, specifically using incidence matrices to represent networks of resistors and batteries. It explains how the null space of an incidence matrix can be used to determine the conditions under which there is no current flow in a circuit, which corresponds to a state of equilibrium where all voltages are equal. The discussion also covers the row space and how it can be used to identify vectors that represent valid potential differences in the circuit. The concept of graph reduction to a tree structure is introduced, explaining how cycles in a graph correspond to dependent rows in the matrix, and how the dimension of the row space relates to the number of edges that can be added without creating loops in the graph.

15:03

🎓 Learning Resources and Conclusion

The final paragraph serves as a conclusion and a call to action for viewers interested in learning more about matrices and linear algebra. It promotes Brilliant.org as a resource for further education, highlighting its courses on linear algebra, graph theory, and differential equations. The paragraph emphasizes the practical applications of these topics, such as the Google page rank algorithm and laser technology. It also mentions a special offer for the first 200 people who sign up through a provided link, offering a discount on the annual premium subscription. The video concludes with a thank you to supporters and a prompt to follow on social media for future content.

Keywords

💡Matrix

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. In the context of the video, matrices are used to represent systems of linear equations and to perform operations that help visualize and solve these systems. The video explains how matrices can be thought of as sets of vectors, which are fundamental in understanding linear combinations and systems of equations.

💡Vector

A vector is a quantity that has both magnitude and direction. In the video, vectors are used to represent the columns or rows of a matrix. The concept is crucial in understanding how matrices can be manipulated to find solutions to systems of equations. The video demonstrates how column vectors can be visualized and used to find scale factors that satisfy a given equation.

💡Linear Equations

Linear equations are equations that represent a straight line when graphed. The video uses 3x3 matrices to represent systems of three linear equations, where each equation can be visualized as a plane in three-dimensional space. The intersection of these planes, if it exists, gives the solution to the system.

💡Null Space

The null space of a matrix is the set of all vectors that, when multiplied by the matrix, result in the zero vector. In the video, the null space is visualized as the intersection of planes when all equations equal zero, representing all possible solutions where there are no potential differences, akin to a circuit with no current flow.

💡Row Space

The row space of a matrix is the set of all possible linear combinations of its row vectors. The video explains that the row space is always perpendicular to the null space and contains the row vectors themselves along with all their linear combinations. This concept is used to determine whether a given vector can be represented as a combination of the rows of a matrix.

💡Column Space

The column space of a matrix is the set of all possible linear combinations of its column vectors. In the video, the column space is discussed in the context of graph theory, where it represents all possible output vectors that can be formed from the edges of a graph, adhering to certain conditions like Kirchhoff's voltage law.

💡Gaussian Elimination

Gaussian elimination is a method for solving systems of linear equations by performing a series of row operations to transform the matrix into a row-echelon form. The video uses this method to simplify the analysis of systems of equations, making it easier to find solutions or determine the dimensions of the row and column spaces.

💡Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In the video, the dot product is used to determine the perpendicularity between vectors, which is crucial in identifying the row space and understanding the relationship between row vectors and the null space.

💡Linearly Dependent

Vectors are linearly dependent if one can be expressed as a linear combination of the others. The video explains that if a set of vectors is linearly dependent, they do not span the entire space and are confined to a subspace, such as a line or a plane. This concept is important for understanding the column space and the solutions to a system of equations.

💡Graph Theory

Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. In the video, graph theory is applied to understand the relationships between nodes and edges in a network, using matrices like the incidence matrix to represent these relationships and analyze the properties of the network.

Highlights

The video introduces matrix manipulation with 3D software and applications not commonly taught in schools.

A matrix can be visualized as a set of vectors, either as column or row vectors.

Matrices can represent systems of equations, where each row corresponds to a linear equation.

Visualizing a matrix as a linear combination of column vectors provides an alternative to solving systems of equations.

The intersection of planes in 3D space can represent the solution to a system of equations.

The null space of a matrix is the set of all solutions when the system of equations equals zero.

Gaussian elimination is a method for solving systems of equations, but the video presents a visual approach to understanding it.

The video demonstrates how changing a matrix can lead to different solutions, such as a line of solutions instead of a single point.

The dot product of a row vector and the null space reveals the perpendicular relationship between them.

The row space of a matrix is the set of all vectors perpendicular to the null space.

The video explains how the dimension of the null space and row space always add up to the dimension of the matrix.

Linear dependence of vectors is demonstrated through the inability to span the entire space, confined to a line or plane.

The column space of a matrix is the set of all possible output vectors that can be formed from the column vectors.

The video connects matrix analysis to graph theory and network analysis, showing practical applications of linear algebra.

An incidence matrix is introduced as a way to represent a network or circuit in terms of nodes and edges.

The null space of an incidence matrix represents the voltage configurations that result in no current flow in a circuit.

The video concludes by emphasizing the importance of understanding the visual and practical aspects of linear algebra beyond textbook explanations.

Transcripts

this video was sponsored by brilliant

every matrix paint some kind of picture

while matrix manipulation or arithmetic

tells us story and that's not just the

one of how boring this can be in school

at least for me the beginning of

matrices was one of my least favorite

parts of math so I won this to at least

show you what this all looks like with

cool 3d software as well as an

application I never learned in school so

here we go when you're given a matrix it

can often be useful to think of it as a

set of vectors I'll be working mostly

with 3x3 matrices and you can think of

these both as a set of three column

vectors or three row vectors we'll look

into each where the column vectors come

in immediately is when we use this

matrix to represent a system of

equations here I'm sure most you know

this gives you three linear equations

for example the first is 1x plus 2y plus

4z equals some b1 and the rest of the

matrix is all the other coefficients but

another way to visualize this same thing

is to write it as a sum or linear

combination of the column vectors where

XY and z are now just scale factors here

the first equation would be 1 times X

plus 2 times y plus 4z equals b1 the

exact same thing so given some system to

solve you can visually think of this two

ways for the first option you say if I

were to graph each of these or in this

case three planes where do they all

intersect because that intersection is

our solution XYZ and in this case it'd

be 1 comma 1 comma 1 now I'm going to

switch to geogebra real quick because

it's better for vectors but the second

option says to instead take the columns

of our matrix and consider them as

vectors then find which scale factors

are needed such that those vectors add

tip-to-tail to get some other vector b1

b2 b3 so instead of an intersection

we're looking for scale factors and in

this case all of them would be 1 just

add the vectors together as they are

thus 1 comma 1 comma 1 is our solution

just like we saw before so we have two

totally different visualizations for the

exact same question

I like using the intersection one when I

have to solve for x y&z but when I'm

asked what are the possible outputs here

for B then I like thinking of vectors

now I'm going to change the matrix just

a bit and also make the B vector all

zeros this then changes the other

equations and now let's go back to the

3d plot here we have the first and third

equation and unless they're parallel two

different planes will always intersect

in a line now if the remaining plane

happens to intersect that same line as

well which it does then we have an

entire set of solutions XY and Z such

that all these equations are zero the

name we give to those solutions is the

null space it's just the intersection of

all your equations when they equal zero

often that solution is just zero comma

zero comma zero but sometimes there's

more here the null space is one

dimensional just a line in 3d space now

on your homework you want it graph three

planes most likely you do something like

Gaussian elimination where you take two

equations multiply one or both by a

constant and cancel out one of the

variables but instead of just

multiplying by negative two immediately

I'm gonna sweep the constant from zero

to negative two and watch what happens

to the resultant function which

currently is just that second graph in

pink

so look you can see when you add any two

of these linear equations regardless of

the scale factor in front their

intersection or the null space in this

case is preserved the new plane just

rotates about that intersection so we

may have a totally different plane here

but we haven't lost the solutions so we

can just replace either equation one or

two and still go through the analysis

but now the arithmetic is a little

easier because one of the coefficients

is 0 if you do the same thing with

equations 1 & 3 then one plane actually

becomes another

this happens because if we replace

equation three these last two planes are

the exact same now that means if we were

to continue the elimination we get a row

of all zeros and four square matrices at

least a single row of zeros means we

have a single free variable this tells

us we have infinitely many solutions to

the system and we say Z can be anything

it's free

but x and y depend on that value so we

don't just have any solution those

dependent variables correspond to

something called pivots and since

there's only one free variable then our

null space will be one dimensional and

by the way if we did have three planes

that only intersect at a single point

then the elimination eventually leads to

a plane of one variable like in this

case Z equals one and from there we

would back solve to get Y and X but

anyways now I want to complete the

picture by putting back the original

equations and graph now what if I told

you that the dot product of the vector 1

comma 2 comma 4 and some random vector X

Y Z is 0

well that means these two vectors are

perpendicular

but look the actual dot product or 1 X

plus 2y plus 4z equals 0 is our first

equation an XYZ represents the null

space that line of solutions so our

equation says the first row vector of

our matrix 1 comma 2 comma 4 is

perpendicular to the null space and the

second equation says the second row

vector is also perpendicular to that

same line and same with the third these

are all just dot products being equal to

0 and the set of all vectors

perpendicular to the null space line is

this plane here and this is what we call

the row space this is always

perpendicular to the null space it

contains the three row vectors all three

are in that plane and it also contains

every linear combination of those row

vectors so we have a one-dimensional

null space and a 2-dimensional row space

which add to 3 and that matches this

dimension of the matrix

just note that this will always be true

but don't forget these equations which

represent planes and now we know also

dot products with the null space can

also be thought of the combination of

the column vectors since this is the

exact same question we already know

there are XYZ solutions to this that sum

to the zero vector it's just all the

values that made up that null space line

from before so there are infinitely many

scale factors that make this work and

when a set of vectors can combine to the

zero vector given scale factors that

aren't all zero then those vectors are

linearly dependent or you can also say

one of these vectors is just a linear

combination of the other two same thing

when you have a square matrix with

linearly dependent vectors it means

those vectors don't span the entire

space therein they're confined to like a

line or in this case a plane all the

column vectors are found here and also

all of their linear combinations all

possible tip-to-tail summations the name

we give to that plane that the vectors

span is the column space see often you

could put any three vector here and find

a solution which would mean the vectors

are linearly independent but in the

dependent case we can't have any

solution the output vector has to lie

within this plane the column space in

order for a solution to exist the column

space and the row space which I'll throw

in here as well usually look very

different but they're always the same

dimension both are 2d in this case for

non square matrices the row and column

space are way different here the column

space is just the XY plane these four

vectors can only combine to some other X

comma Y vector but the row space is the

plane spanned by these two vectors in

four dimensional space however both

those spaces are planes that are

themselves two-dimensional so that

aspect does match but graphically these

are very different now with regards to

elimination the obvious reason as to why

this is important is because it's used

to solve systems of equations

when there are many of those equations

which can come up in circuits or other

physical systems then we might not solve

things by hand but we do have to tell

computers how to get a solution however

there's even more of a picture and story

beyond just solving these equations and

that has to do with graph theory and

networks let's say we have some directed

graph with four nodes and five

connecting edges and I'll actually label

all these edges e1 through five and the

nodes and one through four now you can

think of this like a circuit where the

edges are either resistors or a battery

or whatever where current flows and the

nodes would all have some specific

voltage in fact I'll change the labels

to voltages to say consistent with this

then the arrows would sort of represent

current although we can't know the

direction yet until at least here we

know if the voltage is positive or

negative now we can represent this

network with something called an

incidence matrix that will have four

columns for the four nodes and five rows

for the five edges to fill this in just

consider the first edge on the graph

it's coming out of v1 and going into v2

so we put a negative one under v1 and a

positive one under v2 the rest are zero

since they aren't connected to e1 e2 is

then coming out of v2 and going into v3

so we put a negative one under v2 and a

1 under v3 then zeros for the non

connected notes this is all there is to

it negative ones for the out of nodes

and positive one for the into notes so

the rest of the matrix would look like

this now when we multiply this matrix by

a vector of the voltages it equals every

difference between connected nodes or

really potential differences that's like

the voltage drop across resistor or a

battery so now what does the null space

of this matrix represent will remember

the null space is all the solutions here

or the voltages that output all zeros or

no potential differences which is like

asking which voltages will result in no

current well I'm not going to show it

but using Gaussian elimination we get

this

matrix here which again has the same

null space all we did was rotate the

higher dimensional equations around

their intersection and this matrix has

three pivots and one free variable

this means v4 can be whatever and the

rest of the voltages are dependent on

what we pick I'll say v4 equals some

arbitrary T and since the other

equations are just going to lead to V 4

equals V 3 V 3 equals V 2 and V 2 equals

V 1 then every variable would have to be

T or whatever V 4 was selected this is

our null space just a line in four

dimensions we can pick something for V 4

like ground or 5 volts or whatever and

so long as everything is the same then

we have no potential differences or

really no current yeah it's pretty

obvious if you know your circuits but it

gives you an idea of what the null space

really means here and with regards to

the row space if you were asked whether

some vector is a part of it or it can it

be made by combinations of the rows then

all you gotta do is see if it's

perpendicular to the null space and

doing a dot product we see that it is

since we get out 0 in fact so long as

all these numbers add to 0 then it's

definitely in the row space for this

matrix one thing that did have some more

meaning though is the elimination we did

to reiterate what we have here is the

original incidence matrix on top and the

reduced matrix on bottom the original

graph looked like this but now I'm going

to plot the graph or network associated

with the bottom or reduced incidence

matrix which would give us this here

it's the same graph minus 2 edges but

the thing to realize is that it has no

loops meaning it's a tree and it turns

out this will always be the case every

connected graph reduces to a tree and

certain rows or edges that create loops

like this one that represents this edge

eventually reduce to all zeros so we can

say cycles lead to dependent rows since

they reduced to 0 also the dimension of

the row space or 3 in this case means

you can have three edges in this graph

without any loops but any fourth edge

will create one

lastly the column space is just what all

the columns can combine to or any

possible output vector be from a linear

combination of these vectors if you go

through with the analysis you find the

columns combined to any vector so long

as B 1 plus B 4 minus B 5 equals 0 and B

1 plus B 2 plus B 3 + B 4 equals 0 this

definitely has a physical meaning I'm

using the letter B as a filler but

really B 1 is just the first row

summation so really V 2 minus V 1 B 4 is

V 1 minus V 4 and B 5 is V 2 minus V 4

so these values really just represent

potential differences between two

connected notes and bringing back our

original graph in circuit form we find

those are the voltage drops in this loop

thus the potential differences in this

loop sum to zero and this is a

fundamental law of circuits known as

Kirchhoff's voltage law it emerges from

analyzing the column space of the matrix

and by the way the other equation

corresponds to the larger loop where the

voltages must also sum to zero so if you

were given a vector and had to determine

whether it's in the column space you

just need to see whether it obeys

kerkoff's voltage law this vector does

not cuz this loop fails to sum to zero

for example thus it's not in the column

space everything we've seen here might

not be what you typically learn when it

comes to elimination row and column

spaces and so on but within linear

algebra there's almost always an

interesting picture or story going on

beyond what your textbook is telling you

and if you want to dive deeper into what

we've seen here as well as more advanced

topics you can check out brilliant org

the sponsor of this video to continue

with the applications of matrices and

linear algebra brilliant actually has

several courses to learn from first

their linear algebra course covers all

the basics of matrices but it even gets

to adjacency matrices the use of

matrices in graph theory and unique

applications like the Google page rank

algorithm you can go beyond this though

in their differential equation series

which covers on

damn systems matrix Exponential's and

even more advanced applications like

laser technology and the associated

equations covering this wide range of

applications really does help connect

all the little pieces of linear algebra

from determinants to eigenvectors to

diagonalization and so on so you gain a

much better understanding of the big

picture and as you can see brilliant

courses all come with intuitive

animations and tons of practice problems

so you know you have a solid

understanding of whatever topic in math

science or engineering you're interested

in learning also the first 200 people to

go to brilliant org slash text are or

click the link below will get 20% off

their annual premium subscription and

with that I'm gonna end that video there

thanks as always my supporters on

patreon social media links to follow me

or down below and I'll see you guys in

the next video