What Is A Tensor Lesson #1: Elementary vector spaces

we were going to a pro

tensor is by starting with the concept

of a vector and we're going to begin

from the very very basics and we're

going to clear up how to get from the

concept of a vector to the concept of a

tensor so we're going to start this

lecture with an elementary understanding

of what a vector is and I don't want you

to think that that's going to be

something familiar because in your mind

or in the mind of many students who

approach the subject they think they

know all about vectors because they've

made their bones because they have in

physics and in basic electromagnetism

and mechanics they know how vectors work

they know how to add two vectors

together they know how to take the dot

product between two vectors right they

know how to take the cross product

between two vectors to produce a third

vector they know all kinds of things

about vectors and they're very good with

them you know how to translate them and

move them around they know how to scale

them right that's they know how to they

have a very good understanding of how

vectors function the problem is is all

of that stuff we need to forget we need

to actually delete from our mind because

we are going to start with the

mathematical concept of a vector which

is not the same thing so everything you

know about vectors we erase and we're

going to start fresh and where do we

begin we begin with the notion that a

vector is an element of a set and that

set is called a vector space and I'll

call it V s for vector space and a

vector space is a set and every element

in it is a vector so if you come out of

this vector space say you're out here

the element W or you're the element V or

you're the element s you are a vector

and now since there are many different

types of sets in the world we have to

understand what kind of set makes a

vector space what is it that actually

makes because there are many sets that

you can have it's not just every set as

a vector space you have to have a

certain set of properties associated

with

the set and those properties are what's

going to distinguish a vector space set

from any other set and the first key

property is that it must have in

addition to the set itself it must have

an operation called addition and it's

vector addition the idea between for

vector addition is that with if you put

a vector on the left and vector on the

right you're going to get a vector

result so here we might put W V and

we're going to get another vector out

and we could call it t the vector

addition allows you to add two vectors

together and what's important about it

is that is that it only works for

vectors in the set it's not a general

addition rule that allows you to add

vectors from different vector spaces or

different spaces altogether

it only allows you to take two vectors

in the set know some of these or

whatever still back here you can take

two vectors put on the left and right

and you get a third member and that

member is also in the set so in this

case T would also have to be part of the

vector space because it must be closed

you must be able to add any two vectors

and you look and the one thing that you

get as a result is in the vector space

it's in the vector space itself you

can't do that you don't have a vector

space so you have to define this concept

of addition then the next thing you need

is you need to be able to reach in to a

bucket of numbers and that bucket of

numbers are the bucket of real numbers

all the real numbers live in this little

bucket say and you need to be able to

pull out any real number we'll call it a

and you need to have an a sense of how

to multiply a vector from the vector

space any a vector in the vector space

by this real number and that

multiplication is called scalar

multiplication and so we symbolize that

by the real number times the vector and

that is an element of the vector space

we'll call the vector space here say W

double using the vector space so so any

scalar times

a vector is also a vector in W and this

process here is called scalar

multiplication and the objects that come

out of the real numbers these the real

number bumps bin are called scalars now

vector spaces use this real number bin

if they use the real number bin they are

called real vector spaces it's a real

vector space if it uses a bin of real

numbers if it used a bin of say complex

numbers then it would be called the

complex vector space so you you almost

have to distinguish if you're going to

create a vector space you have to assert

not only this addition property but you

have to make a decision is it going to

be real numbers or complex numbers

obviously the complex numbers includes

the real numbers so but you still have

to choose and for general relativity we

will always always choose real vector

spaces for now there is some complex

vector spaces in general relativity but

not anything we're going to talk about

in these lectures so we don't worry

about complex vector spaces just real

vector spaces so now once we've done

this once we've got our we've got our

addition property we've got our scalar

multiplication property then what we're

going to do is we're going to work on

the combination of the two and this

should be very simple if I take a if and

this is what I'll do I'll show you this

is my vector space right it's the vector

space we're going to call it V it's got

its addition property it's got the real

numbers the scalars from the real

numbers and if I take one vector that's

scaled by a real number and add it to

another vector that's scaled by a real

number and both of these vectors come

from V I should be able to get another

vector in the vector space and this

makes perfect sense of course because

this is a

during the vector space this is a vector

in the vector space this is the addition

property the vector addition property

associated with this vector space

therefore it must be that the sum of

those two is also in the vector space

and once I've asserted this then I just

need to assert the simple point of

linearity where if I did aw plus a t I

get a times W plus T which means which

means that the scalar does the scaled

prata

the scalar product with W plus the

scalar product with T is the same as

adding W plus T and multiplying by the

scalar and this is simply a very

critical property called linearity and

it means that our vector space is linear

and this didn't have to be that way by

the way it could have been that this

equaled say a squared W plus T that does

happen for some exotic forms of spaces

but not the ones we're talking about

this is not what we're using so we've

got this we've got several things we've

got our um our linearity property which

encompasses both our vector addition

property and our scalar multiplication

property and then one last thing that

defines a vector space unambiguously is

we need to make sure that any vector W

that is an element of this vector space

say our vector space is V if W is if if

W is an element of V then there's

another vector in the vector space V

called - W and that is characterized by

the fact that W with a vector addition

of minus W equals zero and sure enough

zero therefore is always a vector in

every vector space zero must be a vector

in the vector space and every vector

must have its opposite and yes the

opposite is if I take from my bin of

real numbers if I take minus one and I

use that to multiply by a vector W that

product

is in fact - W and it's always part of

the vector space so so far so good we've

got we've got our vector space V and

we've got the vector addition property

we've got the scalar multiplication

property from the real numbers so this

is a real vector space and we know that

it's linear and that is our vector space

now an interesting thing is that we have

to be able to answer to one or two

important questions about in elementary

vector space we've already answered one

is it a real vector space or complex

vector space there's actually two other

kinds it could be quatrain yannick or it

could be octi onic but there's only four

there's four different kinds of vector

spaces and and anything other than those

four is a more of a mathematical

generalization of the concept but when

we talk about vector spaces we're almost

always talking about we're almost always

talking about real or complex vector

spaces complex vector spaces are

important in quantum mechanics but in

general relativity we're dealing with

real vector spaces but if I did this

again I could create another vector

space W and it'll also have be a real

vector space and it will have its own

vector addition property now I can pull

out vectors from W say I pulled out well

let's let's say I called it our I pulled

out a vector s and I pulled out a vector

T and from V let's say I pulled out a

vector a little W a little Q and how

about little P right so these are

vectors from W these are vectors from W

these are vectors I'm sorry these are

vectors from W and these are vectors

from Q now the vector addition property

of W is such that I can take any of

these two and add them and I can get

another vector

inside inside V so W plus Q equals say M

likewise I can take R plus s and I can

get another vector out of

out of w and say that one was called

i'll say say it was t right the thing

that's very important to know is this

vector addition property only works for

these vectors in this vector addition

property only works for those vectors

this is not the same plus sign as this

and the only thing that gives it away is

knowing that r and s are elements of W

and W and Q are elements of V if you

didn't know that you might think that

these represent the same operation but

these are different operations you can

never never write W plus R because W

comes from V and R comes from W and

there is no defined operation that adds

elements of V to elements of W it just

doesn't exist we have not defined it now

you could define something like that

there it is possible but that's not what

we're doing we're creating nothing all

we're creating is addition properties

for individual vector spaces so but it

is also now an important question to ask

what's the difference between this

vector space in this vector space other

than the name and they're both real

vector spaces so you could imagine this

is a complex vector space that would be

different from a real vector space but

symbolically or mathematically is there

a way of distinguishing these two and

the answer is often there is not well

there is one key characteristic that can

distinguish between two vector spaces

that's the dimension of the vector space

so the way we learn about dimensions is

we're going to ask the very fundamental

question I draw a random vector any

vector any arbitrary vector out of V out

of this space V let's say we pick them q

if I draw drew an arbitrary vector out

of V I want to know what's the minimum

number of other vectors I would need to

be able to linearly combine them to

create Q so say there's a vector a W

plus B

the P plus C let's say n plus D Oh about

Oh and then we could go on and on and on

and the question is is I need to find a

minimal set of vectors a P and O that

multiplied by real numbers will give me

any Q in the vector space and if I can

find a minimal set of those vectors in

this case the minimal set might be WP N

and O let's say I can find that minimal

set I know that I can express any vector

any vector in V as a linear combination

of these four basis vectors and that's

what these are called these are called

basis vectors and basis vectors they

they are not unique inside the vector

space you can obviously see why they

wouldn't be unique because if W is a

basis vector than a W would also be a

basis vector because you could just

rescale it by choosing another real

number so clearly basis vectors aren't

unique but what is important is the

number of them I need the minimal number

that can capture every vector in the

vector space and in this case I've said

that the minimal number is four and so I

what I'm saying and now is that the

dimension of V equals four and we're

going to use four dimensional four for

all of our work because four is the

dimensions of space time and space time

is what we're going to talk about we're

trying to shoot for general relativity

so we're going to talk about four

dimensional vector spaces but if V is a

dimension of four and I could put that

right here say make a little circle

around it

what about W well if W has the same

number dimensions then W and V are only

different because they're named

differently there's got to be something

to distinguish them so it's got to be

the name but otherwise if they're the

same dimension they're actually so

similar that the difference is between

these two vector spaces is entirely

superficial and we call that isomorphic

two vector spaces are isomorphic if

they're in

if you can establish a one-to-one

correspondence between the two and if

operations in this vector space are in

correspondence to operations in that

vector space we're not going to talk too

much about it but the point is is that

other than the name these two vector

spaces are mathematically very very very

similar and you really have to come up

with ways of distinguishing them okay so

where we're at now is we've covered the

elementary properties that all vector

spaces must have and those elementary

properties are our they must be defined

with a vector addition they must be

defined with a scalar multiplication

generally for real numbers for what

we're going to do they must be linear

and they must have a dimension and in

our case the dimension is 4 now

understand the only operation we have

between two vectors in one vector space

is if V and if W whoops if V and W are

members of the vector space V I can add

them but I can't do anything else notice

we have not discussed this concept this

is a totally different concept remember

dot product between two little pointy

things that we learned in physics that's

an element of the real numbers right we

have not learned how to take two vectors

and turn them into a real number that

this does not exist in a vector space

there's no notion of a cross product in

a vector space by the way a cross

product produces another vector but not

necessarily in the same vector space as

these two so we don't have a notion of a

cross product we don't have a notion of

a magnitude right which remember that

was V dot V right we do not have a

notion of a magnitude or a squared

magnitude I should say that doesn't

exist none of these things exist in real

pure elementary vector spaces all of

this stuff is advanced in a weird way

right it's it's not very complicated but

it is stuff that's added to vector

spaces that make

the more sophisticated than the

elementary vector space very few things

out there in the world are actually

purely elementary vector spaces but but

all vector spaces are in fact elementary

vector spaces at least and if they don't

have these properties they're not vector

spaces at all but they can have other

properties like dot products and cross

products and magnitudes and things like

that but this is what we're going to

start talking about next

but right now understand this is the

core element of a vector space this is

what makes a vector space so our next

lecture is going to be a little bit more

about how to now start building maps

between vector spaces