What Is A Tensor Lesson #1: Elementary vector spaces

XylyXylyX
20 Mar 201618:29

Summary

TLDRThis lecture delves into the foundational concepts of vectors and tensors, starting from scratch by redefining vectors beyond their common physical interpretations. It emphasizes the mathematical construct of a vector as an element of a vector space, outlining properties such as vector addition and scalar multiplication that differentiate a vector space from other sets. The lecture then progresses to explaining the significance of dimensions within vector spaces, touching upon the notions of linearity and isomorphism. By challenging the audience to discard preconceived notions about vectors, it sets the stage for understanding the complexity and beauty of tensor calculus, crucial for fields like general relativity.

Takeaways

  • 💡 The foundational concept starts with redefining vectors from a mathematical perspective, distinct from their common understanding in physics.
  • 📖 A vector is defined as an element of a set within a vector space, emphasizing the abstraction away from physical concepts.
  • ✍️ Vector space properties are crucial: it must allow vector addition and scalar multiplication, following specific rules to qualify as a vector space.
  • ❓ Dimensionality is a key characteristic of vector spaces, determining the minimal set of basis vectors needed to represent any vector in the space.
  • 📈 The lecture introduces the concept of real vector spaces, using real numbers for scalar multiplication, as the focus for general relativity studies.
  • ✔️ Essential operations for vector spaces include vector addition, which must result in another vector within the same space, ensuring closure.
  • ⚡ Scalar multiplication involves combining a vector with a real number, producing another vector within the same space, highlighting the linear structure.
  • 🖥 Linearity and the principle of superposition are underscored as fundamental properties, enabling the construction of vectors through addition and scalar multiplication.
  • ⭕ The absence of operations like dot and cross products in elementary vector spaces is highlighted, distinguishing pure vector spaces from more complex structures.
  • 📚 The discussion prepares for future topics on mapping between vector spaces, indicating a deeper exploration of mathematical structures in relativity.

Q & A

  • What initial misconception about vectors is highlighted in the lecture?

    -The lecture highlights the misconception that many students think they fully understand vectors based on their familiarity with them from physics and basic electromagnetism and mechanics, such as vector addition, dot products, and cross products.

  • Why does the lecture suggest forgetting everything known about vectors?

    -It suggests forgetting everything known about vectors in order to start fresh with the mathematical concept of a vector, which is fundamentally different from the practical applications of vectors commonly taught.

  • What defines a vector space, according to the lecture?

    -A vector space is defined as a set in which every element is a vector. It must have an operation called vector addition, where adding two vectors results in another vector within the same set, indicating closure under addition.

  • What is the significance of scalar multiplication in a vector space?

    -Scalar multiplication, the process of multiplying a vector by a real number (scalar) to produce another vector within the same space, is significant because it along with vector addition, helps define the structure and properties of a vector space.

  • How does the lecture differentiate between real and complex vector spaces?

    -The differentiation is based on the type of numbers used for scalar multiplication. If a vector space uses real numbers, it's a real vector space; if it uses complex numbers, it's a complex vector space.

  • What is linearity in the context of vector spaces?

    -Linearity refers to the property that allows the combination of scalar multiplication and vector addition in such a way that if two vectors are scaled and then added, the result is the same as adding the vectors first and then scaling the result.

  • Why must every vector in a vector space have an opposite?

    -Every vector must have an opposite to ensure that the vector space is closed under addition. This opposite vector, when added to the original vector, results in the zero vector, maintaining the structural integrity of the space.

  • How is the dimension of a vector space determined?

    -The dimension of a vector space is determined by the minimum number of basis vectors needed to linearly combine them to form any vector in the space. This minimal set of vectors captures the essence of the vector space's structure.

  • What does it mean for two vector spaces to be isomorphic?

    -Two vector spaces are isomorphic if there is a one-to-one correspondence between their elements and their operations, meaning they are structurally the same in terms of addition and scalar multiplication, differing only in nomenclature.

  • Why are certain operations like the dot product and magnitude not initially considered part of a vector space?

    -These operations are not part of the fundamental definition of a vector space. They are advanced concepts added to enrich the structure of a vector space, making it more sophisticated than just the basic requirements of vector addition and scalar multiplication.

Outlines

00:00

🧠 Introduction to Vectors and Vector Spaces

This section introduces the fundamental shift from the physical concept of vectors, commonly encountered in physics, to the mathematical concept integral to understanding tensors. The physical operations familiar to students, such as vector addition, dot product, and cross product, are set aside to focus on the mathematical definition of a vector as an element of a set known as a vector space (VS). A vector space is defined by two key properties: the ability to add two vectors within the space (vector addition) and the ability to multiply vectors by real numbers (scalar multiplication), maintaining closure within the set. The narrative emphasizes that these operations are exclusive to vectors within the same vector space, underlining the specificity of vector space operations and the foundational role these concepts play in progressing towards understanding tensors.

05:04

🔢 From Vector Spaces to Complex and Real Vector Spaces

The narrative continues by distinguishing between real and complex vector spaces based on the type of numbers used for scalar multiplication, highlighting the use of real vector spaces for general relativity. The script delves into the mathematical properties that define a vector space: the addition and scalar multiplication operations, which ensure closure and linear combination within the space. A vector space must also include the zero vector and allow for inverse vectors, ensuring every vector has a counterpart that sums to zero. This foundation prepares for the exploration of vector space dimensions, emphasizing the distinction between vector spaces through the concept of dimensionality rather than their operational definitions, which remain consistent across vector spaces.

10:04

🌌 Dimensionality and Basis Vectors in Vector Spaces

Expanding on the concept of dimensionality, this section explains how to determine the dimension of a vector space by finding the minimum number of basis vectors needed to express any vector within the space through linear combination. The narrative clarifies the non-uniqueness of basis vectors but underscores the significance of the minimal set required for complete vector space representation, known as the dimension of the space. Using the dimensionality concept, the text sets the stage for discussing vector spaces in the context of space-time, specifically adopting a four-dimensional perspective for the study of general relativity. This focus on dimensionality serves as a critical step towards understanding complex concepts within physics and mathematics.

15:05

🔀 Distinguishing Vector Spaces and Isomorphism

The final segment addresses the differentiation of vector spaces through the lens of isomorphism, emphasizing that vector spaces of the same dimensionality are fundamentally similar, barring their nomenclature. Isomorphism is defined as the ability to establish a one-to-one correspondence between elements (and operations) of two vector spaces, rendering their differences superficial. The script also highlights that vector spaces are limited to operations within their elements and that advanced concepts like dot products, cross products, and magnitudes are not inherent to basic vector space theory. This distinction sets the groundwork for further exploration of mappings between vector spaces, indicating a transition towards more complex mathematical structures and their applications.

Mindmap

Keywords

💡Vector

A vector, in the context of this video, is defined not just as an entity with magnitude and direction as commonly encountered in physics but as an element of a set known as a vector space. This perspective shifts from the physical representation to a more abstract mathematical concept. For instance, the discussion emphasizes the need to forget the physical applications of vectors (like cross and dot products) and focus on understanding vectors as elements that follow specific rules within a vector space.

💡Vector Space

A vector space is introduced as a set of vectors that adhere to two main operations: vector addition and scalar multiplication. The video elaborates that for a set to qualify as a vector space, it must satisfy certain properties such as closure under addition and scalar multiplication. This foundational concept serves as the building block for discussing more complex ideas such as linear combinations and dimensions of vector spaces.

💡Scalar Multiplication

Scalar multiplication is described as an operation within a vector space where a vector is multiplied by a scalar (a real number), resulting in another vector within the same vector space. This concept is crucial for understanding the structure and behavior of vectors in a vector space, illustrating how vectors can be scaled or modified while remaining within the space.

💡Real Vector Spaces

The video distinguishes real vector spaces by their use of real numbers for scalar multiplication. This classification is pivotal for the discussion on general relativity in the video, where real vector spaces are primarily used. It sets the stage for further exploration of vector spaces in the context of physics and mathematics, highlighting the importance of the type of numbers (real or complex) used in scalar multiplication.

💡Linearity

Linearity is presented as a property of vector spaces, where the sum of two scaled vectors results in another vector within the same space, adhering to the principles of vector addition and scalar multiplication. This property underscores the mathematical structure of vector spaces, emphasizing the predictability and consistency of operations within them.

💡Dimension

Dimension, in the context of this video, refers to the minimum number of basis vectors needed to represent any vector in the space through linear combinations. The discussion of dimensions is integral for distinguishing between different vector spaces, especially when exploring the concept of spacetime in general relativity, which is described as a four-dimensional vector space.

💡Basis Vectors

Basis vectors are defined as a minimal set of vectors from which any vector in the space can be linearly combined to be formed. The video uses this concept to explain how dimensions of a vector space are determined, emphasizing the foundational role of basis vectors in understanding the structure and capabilities of vector spaces.

💡Isomorphic

Isomorphic vector spaces are described as having a one-to-one correspondence between their elements and operations, despite possibly being named or presented differently. This concept is used to illustrate the fundamental similarity between vector spaces of the same dimension, reinforcing the idea that the mathematical structure of vector spaces is more critical than their names or superficial distinctions.

💡Addition Operation

The addition operation in vector spaces is highlighted as a rule allowing the combination of two vectors to produce a third vector within the same space. This operation is fundamental to the structure of vector spaces, demonstrating how elements within the space interact and how the space adheres to the closure property.

💡Scalar

Scalars are identified as the real numbers used in scalar multiplication within vector spaces. The video emphasizes the role of scalars in defining the types of vector spaces (real or complex) and in operations such as scaling vectors. Understanding scalars is crucial for grasping the interaction between numbers and vectors in vector spaces.

Highlights

Introduction to tensors starting with basic vector concepts.

Clearing misconceptions about vectors learned in physics.

Definition of a vector as an element of a vector space.

Explanation of vector spaces and their properties.

The necessity of vector addition for a set to qualify as a vector space.

Scalar multiplication and its role in vector spaces.

Distinction between real vector spaces and complex vector spaces.

Introduction to the concept of linearity in vector spaces.

The requirement of an additive inverse for every vector in a vector space.

The concept of dimensionality in vector spaces.

Illustration of basis vectors and their significance in defining vector spaces.

The isomorphic nature of vector spaces with the same dimensionality.

Clarification that vector spaces fundamentally only support addition within the same space.

Elaboration on the absence of dot product, cross product, and magnitude in pure vector spaces.

Transitioning from basic vector space properties to mapping between vector spaces.

Transcripts

00:07

we were going to a pro

00:10

tensor is by starting with the concept

00:13

of a vector and we're going to begin

00:18

from the very very basics and we're

00:20

going to clear up how to get from the

00:22

concept of a vector to the concept of a

00:25

tensor so we're going to start this

00:29

lecture with an elementary understanding

00:31

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

00:33

to think that that's going to be

00:34

something familiar because in your mind

00:37

or in the mind of many students who

00:39

approach the subject they think they

00:40

know all about vectors because they've

00:42

made their bones because they have in

00:44

physics and in basic electromagnetism

00:47

and mechanics they know how vectors work

00:49

they know how to add two vectors

00:50

together they know how to take the dot

00:53

product between two vectors right they

00:55

know how to take the cross product

00:57

between two vectors to produce a third

00:59

vector they know all kinds of things

01:02

about vectors and they're very good with

01:03

them you know how to translate them and

01:04

move them around they know how to scale

01:06

them right that's they know how to they

01:10

have a very good understanding of how

01:11

vectors function the problem is is all

01:14

of that stuff we need to forget we need

01:17

to actually delete from our mind because

01:21

we are going to start with the

01:23

mathematical concept of a vector which

01:25

is not the same thing so everything you

01:28

know about vectors we erase and we're

01:29

going to start fresh and where do we

01:32

begin we begin with the notion that a

01:35

vector is an element of a set and that

01:40

set is called a vector space and I'll

01:43

call it V s for vector space and a

01:46

vector space is a set and every element

01:49

in it is a vector so if you come out of

01:52

this vector space say you're out here

01:54

the element W or you're the element V or

01:58

you're the element s you are a vector

02:01

and now since there are many different

02:04

types of sets in the world we have to

02:06

understand what kind of set makes a

02:09

vector space what is it that actually

02:11

makes because there are many sets that

02:16

you can have it's not just every set as

02:18

a vector space you have to have a

02:19

certain set of properties associated

02:21

with

02:23

the set and those properties are what's

02:25

going to distinguish a vector space set

02:27

from any other set and the first key

02:29

property is that it must have in

02:31

addition to the set itself it must have

02:33

an operation called addition and it's

02:37

vector addition the idea between for

02:41

vector addition is that with if you put

02:43

a vector on the left and vector on the

02:45

right you're going to get a vector

02:47

result so here we might put W V and

02:52

we're going to get another vector out

02:53

and we could call it t the vector

02:55

addition allows you to add two vectors

02:58

together and what's important about it

03:00

is that is that it only works for

03:05

vectors in the set it's not a general

03:08

addition rule that allows you to add

03:09

vectors from different vector spaces or

03:12

different spaces altogether

03:13

it only allows you to take two vectors

03:15

in the set know some of these or

03:18

whatever still back here you can take

03:20

two vectors put on the left and right

03:21

and you get a third member and that

03:23

member is also in the set so in this

03:25

case T would also have to be part of the

03:28

vector space because it must be closed

03:30

you must be able to add any two vectors

03:32

and you look and the one thing that you

03:34

get as a result is in the vector space

03:36

it's in the vector space itself you

03:39

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

03:41

space so you have to define this concept

03:43

of addition then the next thing you need

03:46

is you need to be able to reach in to a

03:49

bucket of numbers and that bucket of

03:53

numbers are the bucket of real numbers

03:55

all the real numbers live in this little

03:58

bucket say and you need to be able to

04:00

pull out any real number we'll call it a

04:03

and you need to have an a sense of how

04:06

to multiply a vector from the vector

04:09

space any a vector in the vector space

04:11

by this real number and that

04:14

multiplication is called scalar

04:15

multiplication and so we symbolize that

04:19

by the real number times the vector and

04:22

that is an element of the vector space

04:25

we'll call the vector space here say W

04:29

double using the vector space so so any

04:33

scalar times

04:35

a vector is also a vector in W and this

04:39

process here is called scalar

04:41

multiplication and the objects that come

04:44

out of the real numbers these the real

04:47

number bumps bin are called scalars now

04:52

vector spaces use this real number bin

04:54

if they use the real number bin they are

04:56

called real vector spaces it's a real

05:03

vector space if it uses a bin of real

05:05

numbers if it used a bin of say complex

05:09

numbers then it would be called the

05:11

complex vector space so you you almost

05:20

have to distinguish if you're going to

05:21

create a vector space you have to assert

05:24

not only this addition property but you

05:26

have to make a decision is it going to

05:28

be real numbers or complex numbers

05:30

obviously the complex numbers includes

05:32

the real numbers so but you still have

05:34

to choose and for general relativity we

05:37

will always always choose real vector

05:41

spaces for now there is some complex

05:44

vector spaces in general relativity but

05:46

not anything we're going to talk about

05:48

in these lectures so we don't worry

05:50

about complex vector spaces just real

05:52

vector spaces so now once we've done

05:55

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

05:58

addition property we've got our scalar

05:59

multiplication property then what we're

06:02

going to do is we're going to work on

06:08

the combination of the two and this

06:11

should be very simple if I take a if and

06:14

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

06:16

is my vector space right it's the vector

06:18

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

06:20

its addition property it's got the real

06:23

numbers the scalars from the real

06:26

numbers and if I take one vector that's

06:32

scaled by a real number and add it to

06:35

another vector that's scaled by a real

06:37

number and both of these vectors come

06:38

from V I should be able to get another

06:43

vector in the vector space and this

06:44

makes perfect sense of course because

06:46

this is a

06:47

during the vector space this is a vector

06:49

in the vector space this is the addition

06:51

property the vector addition property

06:53

associated with this vector space

06:55

therefore it must be that the sum of

06:58

those two is also in the vector space

07:00

and once I've asserted this then I just

07:05

need to assert the simple point of

07:07

linearity where if I did aw plus a t I

07:13

get a times W plus T which means which

07:21

means that the scalar does the scaled

07:24

prata

07:25

the scalar product with W plus the

07:27

scalar product with T is the same as

07:29

adding W plus T and multiplying by the

07:32

scalar and this is simply a very

07:35

critical property called linearity and

07:37

it means that our vector space is linear

07:40

and this didn't have to be that way by

07:43

the way it could have been that this

07:44

equaled say a squared W plus T that does

07:48

happen for some exotic forms of spaces

07:51

but not the ones we're talking about

07:53

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

07:55

got this we've got several things we've

07:57

got our um our linearity property which

08:02

encompasses both our vector addition

08:04

property and our scalar multiplication

08:06

property and then one last thing that

08:08

defines a vector space unambiguously is

08:10

we need to make sure that any vector W

08:13

that is an element of this vector space

08:15

say our vector space is V if W is if if

08:20

W is an element of V then there's

08:24

another vector in the vector space V

08:26

called - W and that is characterized by

08:30

the fact that W with a vector addition

08:32

of minus W equals zero and sure enough

08:39

zero therefore is always a vector in

08:42

every vector space zero must be a vector

08:44

in the vector space and every vector

08:46

must have its opposite and yes the

08:49

opposite is if I take from my bin of

08:52

real numbers if I take minus one and I

08:56

use that to multiply by a vector W that

08:59

product

09:01

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

09:05

the vector space so so far so good we've

09:10

got we've got our vector space V and

09:15

we've got the vector addition property

09:17

we've got the scalar multiplication

09:19

property from the real numbers so this

09:20

is a real vector space and we know that

09:23

it's linear and that is our vector space

09:26

now an interesting thing is that we have

09:32

to be able to answer to one or two

09:34

important questions about in elementary

09:36

vector space we've already answered one

09:38

is it a real vector space or complex

09:40

vector space there's actually two other

09:42

kinds it could be quatrain yannick or it

09:44

could be octi onic but there's only four

09:47

there's four different kinds of vector

09:48

spaces and and anything other than those

09:51

four is a more of a mathematical

09:53

generalization of the concept but when

09:55

we talk about vector spaces we're almost

09:56

always talking about we're almost always

10:01

talking about real or complex vector

10:04

spaces complex vector spaces are

10:05

important in quantum mechanics but in

10:07

general relativity we're dealing with

10:09

real vector spaces but if I did this

10:11

again I could create another vector

10:13

space W and it'll also have be a real

10:19

vector space and it will have its own

10:21

vector addition property now I can pull

10:24

out vectors from W say I pulled out well

10:27

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

10:30

out a vector s and I pulled out a vector

10:32

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

10:34

vector a little W a little Q and how

10:39

about little P right so these are

10:42

vectors from W these are vectors from W

10:45

these are vectors I'm sorry these are

10:47

vectors from W and these are vectors

10:48

from Q now the vector addition property

10:51

of W is such that I can take any of

10:55

these two and add them and I can get

10:58

another vector

10:59

inside inside V so W plus Q equals say M

11:06

likewise I can take R plus s and I can

11:11

get another vector out of

11:13

out of w and say that one was called

11:16

i'll say say it was t right the thing

11:19

that's very important to know is this

11:21

vector addition property only works for

11:23

these vectors in this vector addition

11:24

property only works for those vectors

11:26

this is not the same plus sign as this

11:29

and the only thing that gives it away is

11:31

knowing that r and s are elements of W

11:33

and W and Q are elements of V if you

11:37

didn't know that you might think that

11:39

these represent the same operation but

11:41

these are different operations you can

11:43

never never write W plus R because W

11:50

comes from V and R comes from W and

11:54

there is no defined operation that adds

11:57

elements of V to elements of W it just

12:00

doesn't exist we have not defined it now

12:03

you could define something like that

12:05

there it is possible but that's not what

12:07

we're doing we're creating nothing all

12:10

we're creating is addition properties

12:12

for individual vector spaces so but it

12:16

is also now an important question to ask

12:18

what's the difference between this

12:19

vector space in this vector space other

12:21

than the name and they're both real

12:24

vector spaces so you could imagine this

12:26

is a complex vector space that would be

12:28

different from a real vector space but

12:30

symbolically or mathematically is there

12:34

a way of distinguishing these two and

12:36

the answer is often there is not well

12:40

there is one key characteristic that can

12:42

distinguish between two vector spaces

12:44

that's the dimension of the vector space

12:46

so the way we learn about dimensions is

12:51

we're going to ask the very fundamental

12:54

question I draw a random vector any

12:57

vector any arbitrary vector out of V out

13:00

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

13:03

if I draw drew an arbitrary vector out

13:07

of V I want to know what's the minimum

13:10

number of other vectors I would need to

13:13

be able to linearly combine them to

13:15

create Q so say there's a vector a W

13:20

plus B

13:22

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

13:36

Oh and then we could go on and on and on

13:38

and the question is is I need to find a

13:41

minimal set of vectors a P and O that

13:45

multiplied by real numbers will give me

13:48

any Q in the vector space and if I can

13:51

find a minimal set of those vectors in

13:53

this case the minimal set might be WP N

13:57

and O let's say I can find that minimal

14:02

set I know that I can express any vector

14:04

any vector in V as a linear combination

14:08

of these four basis vectors and that's

14:12

what these are called these are called

14:13

basis vectors and basis vectors they

14:16

they are not unique inside the vector

14:19

space you can obviously see why they

14:22

wouldn't be unique because if W is a

14:24

basis vector than a W would also be a

14:26

basis vector because you could just

14:28

rescale it by choosing another real

14:30

number so clearly basis vectors aren't

14:32

unique but what is important is the

14:34

number of them I need the minimal number

14:36

that can capture every vector in the

14:38

vector space and in this case I've said

14:41

that the minimal number is four and so I

14:44

what I'm saying and now is that the

14:45

dimension of V equals four and we're

14:49

going to use four dimensional four for

14:51

all of our work because four is the

14:54

dimensions of space time and space time

14:56

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

14:59

trying to shoot for general relativity

15:00

so we're going to talk about four

15:02

dimensional vector spaces but if V is a

15:04

dimension of four and I could put that

15:06

right here say make a little circle

15:08

around it

15:08

what about W well if W has the same

15:10

number dimensions then W and V are only

15:14

different because they're named

15:15

differently there's got to be something

15:17

to distinguish them so it's got to be

15:19

the name but otherwise if they're the

15:22

same dimension they're actually so

15:24

similar that the difference is between

15:26

these two vector spaces is entirely

15:28

superficial and we call that isomorphic

15:33

two vector spaces are isomorphic if

15:36

they're in

15:36

if you can establish a one-to-one

15:38

correspondence between the two and if

15:40

operations in this vector space are in

15:42

correspondence to operations in that

15:44

vector space we're not going to talk too

15:45

much about it but the point is is that

15:47

other than the name these two vector

15:49

spaces are mathematically very very very

15:51

similar and you really have to come up

15:55

with ways of distinguishing them okay so

15:58

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

16:01

elementary properties that all vector

16:03

spaces must have and those elementary

16:06

properties are our they must be defined

16:10

with a vector addition they must be

16:12

defined with a scalar multiplication

16:13

generally for real numbers for what

16:15

we're going to do they must be linear

16:17

and they must have a dimension and in

16:24

our case the dimension is 4 now

16:29

understand the only operation we have

16:32

between two vectors in one vector space

16:35

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

16:46

members of the vector space V I can add

16:50

them but I can't do anything else notice

16:53

we have not discussed this concept this

16:56

is a totally different concept remember

16:58

dot product between two little pointy

16:59

things that we learned in physics that's

17:01

an element of the real numbers right we

17:04

have not learned how to take two vectors

17:06

and turn them into a real number that

17:07

this does not exist in a vector space

17:09

there's no notion of a cross product in

17:12

a vector space by the way a cross

17:14

product produces another vector but not

17:16

necessarily in the same vector space as

17:19

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

17:23

cross product we don't have a notion of

17:25

a magnitude right which remember that

17:27

was V dot V right we do not have a

17:29

notion of a magnitude or a squared

17:33

magnitude I should say that doesn't

17:35

exist none of these things exist in real

17:37

pure elementary vector spaces all of

17:40

this stuff is advanced in a weird way

17:43

right it's it's not very complicated but

17:46

it is stuff that's added to vector

17:49

spaces that make

17:49

the more sophisticated than the

17:51

elementary vector space very few things

17:53

out there in the world are actually

17:54

purely elementary vector spaces but but

17:58

all vector spaces are in fact elementary

18:01

vector spaces at least and if they don't

18:02

have these properties they're not vector

18:04

spaces at all but they can have other

18:06

properties like dot products and cross

18:08

products and magnitudes and things like

18:10

that but this is what we're going to

18:12

start talking about next

18:13

but right now understand this is the

18:16

core element of a vector space this is

18:18

what makes a vector space so our next