Dear linear algebra students, This is what matrices (and matrix manipulation) really look like
Summary
TLDRThis 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
📊 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.
🔍 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.
🔌 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.
🎓 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.
Mindmap
Keywords
💡Matrix
💡Vector
💡Linear Equations
💡Null Space
💡Row Space
💡Column Space
💡Gaussian Elimination
💡Dot Product
💡Linearly Dependent
💡Graph Theory
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
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