In this illuminating lecture by Dr. Aliza Malek from the Technion Institute, the intricacies of vector spaces are explored through foundational axioms in linear algebra. Dr. Malek revisits the definition of a vector space, emphasizing its ten core properties. This session focuses on enhancing those properties with five additional axioms, enriching our understanding of vector space theory. The lecture systematically proves each axiom, fostering a deeper comprehension of concepts like uniqueness of addition-indifferent elements, vectors' opposites, and conditions under which scalars and vectors equal zero. These axioms are vital for proving more advanced concepts in vector spaces, setting a robust groundwork for future exploration in linear algebra.
Highlights
Chapter 4 delves into vector spaces, with Dr. Aliza Malek leading an engaging session on foundational axioms 🔍.
The lecture revisits the definition of vector spaces and showcases standard versus non-standard examples for a diversified understanding 🌈.
A key focus is on proving the uniqueness of addition-indifferent elements and vectors' opposites to ensure consistency within vector spaces 🧩.
Dr. Malek demonstrates proofs of five new axioms that extend the ten defining properties of vector spaces, covering scalar-vector relationships and more 🎓.
The session concludes with a solid foundation laid for tackling more advanced topics in vector spaces, emphasizing the educational value provided by these axioms 📚.
Key Takeaways
Vector spaces are defined by ten core properties, forming the backbone of the course 🤓.
Dr. Malek introduces five new axioms that build upon these properties ✔️.
Each axiom is thoroughly proven, reinforcing the foundational understanding of vector spaces 👩🏫.
Uniqueness plays a critical role, from addition-indifferent elements to vectors' opposites, ensuring consistency 🌟.
The relationship between scalars, vectors, and zero is pivotal, with 'zero' acting as a unique transformative element in vector spaces ⚖️.
Overview
In this engaging session led by Dr. Aliza Malek, vector spaces are explored with precision and clarity. The lecture revisits the fundamental definition of vector spaces, highlighting ten core properties that form the bedrock of linear algebra. Dr. Malek introduces five additional axioms to enrich these foundational principles, setting the stage for a deeper exploration into the subject.
The heart of the lecture lies in demonstrating the proofs of these new axioms. Each axiom is meticulously proved, emphasizing critical elements such as the uniqueness of the addition-indifferent element, the uniqueness of opposites in vectors, and the pivotal role of zero in scalar-vector equations. These proofs not only solidify the understanding of vector spaces but also prepare students for advanced topics in the field.
Concluding the session, Dr. Malek ensures that students are well-equipped with both ten properties and five axioms now available at their disposal. This comprehensive grounding empowers students to explore more complex concepts within vector spaces, making this lecture a critical resource for anyone delving into linear algebra.
Chapters
00:00 - 30:00: Chapter 4 - Vector Spaces Chapter 4 covers the concept of vector spaces, exploring both standard and non-standard examples. The chapter begins with a review of the definition of a vector space, which involves a group V with addition and a field F for scalar multiplication. It sets the foundation for understanding the theory related to vector spaces within linear algebra.
124 - מרחב וקטורי: משפט בסיסי Transcription
00:00 - 00:30 Chapter 4 - Vector Spaces
Vector Space - Basic Axiom Linear Algebra Course
Dr. Aliza Malek Hello, thank you for coming back. We have already seen
seen what a vector space is, and we have seen standard
and non-standard examples. The standard ones will be used later on,
the non-standard ones were for seeing that we can also define
things that are a little weird. And now it's time for the theory,
so let's get started. First of all, as usual,
we recall the definition. I have the group V with addition, I have the field F that provides us
with scalars to multiply by a scalar,
00:30 - 01:00 and we have a vector space
if all ten properties hold. What we will do now is we will define,
formulate, and prove a basic axiom. an axiom that will enrich these
properties a little more for us, so that when we have a vector space,
in addition to all these ten properties, we will have five more
additional properties which will stem out of the
axiom we will now prove. So here is the axiom: Let V be a vector
space over the field F.
01:00 - 01:30 If you ask me,
I recommend you prepare a stamp. From this moment on, every axiom
and every definition will begin the same: Let V be a vector space over F. We will get situated
inside a vector space, and will begin to define concepts,
formulate axioms, and prove them. So here is the first
axiom with five clauses: The first clause says,
the addition-indifferent in V is unique.
01:30 - 02:00 Remember, the property says:
"exists within V, addition-indifferent". It doesn't say how
many there are. Here we prove that if
we have already proved that we have a vector space,
the addition-indifferent must be unique, there is only one as such. The opposite of v in V
is unique as well. Do you have a vector? It must have an opposite one. We didn't say how many,
the axiom will state: "There is always only one". Note that if you
ever want to check
02:00 - 02:30 whether you have a
group that is a vector space, and you discovered two terms
that are addition-indifferent, you can immediately say that
it will not be a vector space. But, when proving a vector space,
there will be no need to prove uniqueness. If there will be ten properties, using the ten
properties I will prove that is addition-indifferent
is unique, and that the opposite
is unique, let's continue. For every v in V
there exists 0v=0. Many zeros, it's confusing indeed.
02:30 - 03:00 So let me help you read this. I don't always help to read,
sometimes I just ask, what is this zero,
and what is that zero? Since this is the first time,
I will help you read this. How do I help? I'm just anotate, here: The zero that is
written here is 0F, which means that it is
the scalars addition-indifferent in F, times any vector v,
no matter which one, we get 0V,
03:00 - 03:30 The vectors addition-indifferent
in the group V. A zero scalar multiplied
by some vector v, gives the zero vector,
that's what it says here. For every v in V there exists: -1 times v equals -v. What does it say here? That if you take the opposite
to the multiplication-indifferent, and multiply the vector v by it,
you will get the opposite of v.
03:30 - 04:00 The opposite of the indifferent,
multiplied by v, gives the opposite of v. And the last clause says that
if alpha times v equals zero, then alpha = 0, or v = 0, but the opposite is also true,
because it's an "if and only if" statement. Alpha equals zero,
or v equals zero, means
04:00 - 04:30 that alpha times v equals zero. A scalar times a vector will
give the zero vector, if and only if, at least one of them,
either the scalar or the vector, is zero. We have no other way to get the product of a scalar with a vector,
and get the zero vector. We will prove all five sections. So here is the proof: A, the addition-indifferent
is unique, let's prove it. Let's assume by negation that there
are two addition-indifferent terms in V,
04:30 - 05:00 we will call them e1, and e2. If e1 is addition-indifferent,
it means that e1 plus e2, is e2. if e1 indifferent,
we get the other one. Wait, but e2 is also indifferent,
so e1 plus e2 will actually give me e1. e1 plus e2 on one side,
e1 plus e2 on one side, are equal, and therefore the
right side must also be equal.
05:00 - 05:30 In other words, if the left side
is equal, so is the right side, therefore, e1 must be equal to e2. It looks familiar? Indeed, this proof is very familiar, we saw it when we proved the uniqueness
of the addition-indifferent in the field, it's exactly the same proof, only there e1 and e2
were scalars in the field, This time e1 and e2
are vectors in V. Which are actually the
same vector, because e1=e2,
05:30 - 06:00 and it is the addition-indifferent
which we denote by zero. The opposite of v, that is, -v,
which exists according to five, is unique. -v, we already have. If I would like to think for a
moment that I have another one, I will come to the conclusion
that it is the same one. So let's assume again by negation
that v has another opposite, let's call it v1, for instance.
06:00 - 06:30 So I know that v+v1=0. What is this zero then? That's right, it's the zero of v. We add vectors, we get a vector. Let's add the opposite of v, that because of property
five we know we have, on both sides of the equation. What do we get? Here is -v that I added to v+v1,
and here is -v that I added to zero. Now we will use the
properties of a vector space.
06:30 - 07:00 For example, we know
that these brackets can be moved to -v plus v,
and it's worth my while, because I know what
-v plus v equals to. I also know that
-v plus zero will give me -v, because each vector to which
the addition-indifferent is added, gives the original vector. And so, we will get... First we will move the brackets. -v plus v is a vector,
plus the opposite, will give me the zero,
07:00 - 07:30 and here,
I will have a zero as well. And so, we get 0 plus v1, 0 plus v1, on one side, and on the other side,
-v plus 0, that is the indifferent, is -v. But again we have zero plus v1. Zero plus v1, is v1. And so we get that
v1 is actually -v. Zero plus v1, is v1, equals to -v.
07:30 - 08:00 I had an opposite -v, I thought
I had another opposite which I called v1, and I came to the conclusion
that they are both the same, so we only have
one opposite term. Section C, for every vector
there exists: 0*v=0. I remind you that zero
scalar times vector v, gives the zero of v,
the zero vector. Sometimes it's a bit confusing,
but if we know that this is v,
08:00 - 08:30 that zero must be a scalar, because we don't
multiply vectors. And the result must be within V,
so we must get a vector. So zero belongs to F,
that is, this zero, within the context I understand that
we are talking about the zero of the field. Meaning, it's the scalar. It is addition-indifferent,
we already know that. That's why we work with a field we
have already learned a long time ago,
08:30 - 09:00 and we know that zero
is addition-indifferent. Therefore, we can
write the following: The scalar zero
times the vector v, instead of the scalar zero, we will
simply write zero plus zero, times v. Now, we will use property
seven, which states, that we are allowed
to open the brackets. This is zero times v,
plus zero times v. Let's see what we got... I take the beginning
and the end, and compare.
09:00 - 09:30 Zero times v equals 0*v+0*v. What is zero times v? We don't know of course, this is
exactly what we are trying to prove. But we know it's within V. And if it is within V, because of the
closure for multiplication by a scalar, we know it has an opposite,
which is the opposite of 0v? Very simple, -0v, here it is:
09:30 - 10:00 And we will add the -0v to both sides of
the equation, and let's see what we get. -0v plus 0v, I added to one side, and -0v on the other side,
plus what I had, 0v plus 0v. Of course now we will use
our ability to move the brackets from 0v plus 0v, to -0v plus 0v. Let's do this, move the brackets.
10:00 - 10:30 Now we will use the
known properties again. A term plus the
opposite gives me zero. A term plus the
opposite gives me zero, and so, we get
zero equals zero, plus 0v. Once again I have
zero plus something. Zero plus something is something, and so we got that
zero is equal to 0v.
10:30 - 11:00 We took a scalar, multiplied it by
a vector, and got the zero vector. Here, again I helped you
to anotate, do not forget to do it. A scalar multiplied by a vector
gives the zero vector. Section D, for every V there exists: The opposite of the
multiplication-indifferent times the vector is the opposite of the vector.
11:00 - 11:30 How ill we do it? We will calculate v plus -1 times v. Let's see what is the result,
let's do the math. v plus -1 times v, equals: Here is v, and here is v, we would very much like to
take v outside of the brackets, but I will have
nothing left over here: In order to have something left,
I have to use property nine, which states that instead of v,
we can simply write 1 times v.
11:30 - 12:00 How lucky we are
to have property nine. Now, v appears both in the
first and second added terms, therefore, according
to property seven, we can take it out of
the brackets, and get: The first scalar plus the second scalar,
which is 1 plus -1, times the vector v. Wait, a term plus the opposite
is the zero scalar in the field, and so we get zero times v.
12:00 - 12:30 And we just proved
that the zero scalar multiplied by some vector v,
gives the zero vector. So what happened here? I got that when I take v, and I add -1 times v to it,
I get the addition-indifferent term. This means that -1 times v
should be the opposite of v.
12:30 - 13:00 Wait, but which is
the opposite of v? The opposite of v is
denoted to be -1 times v, and how many opposites
did we prove there are? In section B we saw that there
is only one such, and therefore, whichever term here plays the role of
the opposite, must be the opposite itself. In other words, -1 times v,
which behaved like the opposite, must be the opposite,
because there is only one like it. Therefore, -1 times v has no choice
but to be the opposite of v, which is -v.
13:00 - 13:30 Section E, alpha times v
equals zero, if and only if, alpha equals zero,
or v equals zero. We have two
directions to prove here. Let's start from the
other direction. What does other direction mean? Well, since I start from the left side,
and reach the right side. Let's say that alpha
equals 0, or v equals 0,
13:30 - 14:00 and we would like to prove
that alpha times v equals zero. If alpha equals zero,
what do we get? According to section C, I will write zero here,
and what will I get? Alpha times v
equals zero times v, equals zero. I already know it's zero, why? We proved it in section C.
14:00 - 14:30 What happens if v equals zero? Now, alpha v should be
replaced by alpha times zero. That we haven't proven yet. But, we will reproduce
the exact same proof. What we did in section C with
the scalar, we will do the same thing with the vector
instead of the scalar. What does that mean? Take a look: What did we do in section C? We took the scalar 0,
multiplied it by v,
14:30 - 15:00 and we split 0 into 0 plus 0. We multiplied by v,
from there we flowed with it. What did we get in the end? 0F times v, equals 0v. What will we do now? We will take alpha times zero, we will divide this zero
into zero plus zero, only this time,
it will be 0v plus 0v. Here we wrote the scalar
as 0 scalar plus 0 scalar,
15:00 - 15:30 here we will write the zero vector
as zero vector plus zero vector. We will open brackets, take the
beginning, take the end, add the opposite, recreate exactly the same proof. And what will
come out in the end? That any alpha scalar
times the zero vector, gives us the zero vector. A recreation of the proof. Again, I suggest that
you solve it by yourself.
15:30 - 16:00 Now we return to section E,
to prove the first direction. The other direction
is in the proof, but in terms of the axiom
it is the first direction. It is given that alpha
times v equals zero, I want to prove that either
alpha is zero, or v is zero. So let's assume... Pay attention to the zeros, 0 of V, 0 of F, because it's
the scalar, and v is a zero of V.
16:00 - 16:30 The product alpha times v
is the zero vector, alpha equals zero
means the scalar is zero, or v equals zero
means the vector is zero. So let's assume that alpha
equals zero, and we are done, why? Because it was given that
either alpha is zero, or v is zero, so alpha is zero,
and we are done. Wait, if alpha is not zero? Not a problem, alpha is a
scalar in the field, it is not zero, therefore it must have an
opposite in the field, alpha -1.
16:30 - 17:00 Let's multiply both sides of our
equation by alpha and -1. I remind you, this is the equation
because this is what is given, and we multiply by alpha -1. Alpha -1 times alpha v,
alpha -1 times 0v. Let's see what comes out. According to property eight, I can move the brackets from the scalar times a vector,
to the scalar times a scalar.
17:00 - 17:30 So I will get
alpha -1 times alpha. First we multiply the scalars,
and then we multiply by v. But alpha -1 times
alpha is 1, the 1 of the field. So I have 1v
equal to alpha -1 times 0v. But what is 1 times v equal to? Remember property nine? Property nine states
that 1 times v is v. And so we got that v is equal
to to alpha -1 times 0v.
17:30 - 18:00 Wait, but we saw that it doesn't
matter which scalar we take, a scalar times the zero vector
is always the zero vector, and so for every
beta scalar we get, that v is equal to alpha -1 times 0,
and the result must be zero. In other words, what did we get? That v must be zero. And that's it, we are done
proving the basic axiom.
18:00 - 18:30 Now, whenever we say:
"Let V be a vector space over F", we will have all ten
properties at our disposal, as well as all five
axioms we proved here, and will be use them to prove much
more complicated and complex things in the vector spaces. That's it for now,
thank you very much.