Full transcript
0:00Today is all about vector spaces . What
0:03a drag . To get what's going on in this
0:11video , you definitely need to know how
0:13groups , functions , and fields work ,
0:15because all this crazy stuff is built
0:17on them . A quick , quick recap on groups
0:20, because vector spaces are later
0:22defined using them . A group is a set
0:25with an operation . The operation maps
0:28two set elements to another set element
0:30. This is also called closure .
0:33Furthermore , the operation is
0:34associative . There is also a neutral
0:37element for the operation . And finally ,
0:39every element in the set has an inverse
0:42. If you can swap the order , the
0:44operation is commutative and the group
0:47is abelian . Sure , you all remembered
0:49that , right ? If that's completely
0:51foreign , just watch our video on groups
0:53. Do you all still remember what a
0:55function is , too ? A function maps from
0:57the domain X to the codomain Y. Every x
1:01from the domain is mapped to the
1:03function value f ( x ) . And that is an
1:06element of the codomain . You'll see why
1:08that's important now . But because it
1:11would be mega bone-dry without an
1:12example , we'll do one in parallel . You
1:15take a set V and then two operations . A
1:18plus with a circle around it and a dot
1:20with a circle around it . In addition to
1:22the vector space , you always need a
1:23field . Every vector space is ,
1:25importantly in math-speak , built over a
1:27field . The set V contains all the
1:30vectors , and to know the field , you
1:32call the vector space — wait for it — a
1:35K-vector space over the field K. Sounds
1:37stupid , and it is . That's why we’ll
1:40use the standard example you all know :
1:42three-dimensional space . For set V , we
1:45use R³ . For the field , we simply use
1:48the real numbers . Every vector in there
1:50has three components , which can be any
1:53real numbers . These are the normal
1:55things you know from school . So , what's
1:58up with the circled parts ? You take the
2:00plus with the circle and define a first
2:03mapping with it . To do this , you first
2:05form the Cartesian product of the set V
2:07with itself . If you have no clue what
2:10that is , just watch our videos on the
2:12Cartesian product . Now you have a set V
2:15cross V. Each item in the set is a
2:17bracket with two vectors in it . Then
2:20you use the plus to map the two vectors
2:23to their sum . Because these are vectors
2:26, the plus with the circle is called
2:28vector addition . Let’s look at this
2:30in our familiar vector space from
2:31school . You grab two vectors , for
2:34example 1 2 3 and 4 5 6 , and write the
2:37plus between them . In our example , we
2:40define our plus with a circle , as you
2:42know from school , adding the components
2:44together with the normal plus . You
2:47might be wondering about one thing
2:49right now : why do we even write a
2:50circle around the plain old vector
2:52addition ? They don't do that in school ,
2:55probably because they're lazy . We
2:57continue to do it so you can see the
2:59difference between vector addition and
3:01numerical addition . So , let's continue
3:03with the multiplication and the circle ,
3:05so that we have a similar second
3:07mapping defined . This is where your
3:09field becomes important for the vector
3:10space . This time , you form the
3:12Cartesian product of the field and the
3:14set of vectors . You then have a bunch
3:16of brackets containing a field element
3:18and a vector . They also call such a
3:20field element a scalar when talking
3:22about vector spaces . And these are now
3:24mapped to the product of both using the
3:26multiplication . In math-speak , K is a
3:29scalar , V is a vector , and you multiply
3:31V by K. That's why this multiplication
3:33is called scalar multiplication . Let's
3:37do it again with our example using the
3:39vectors from school . This time , you
3:42grab a scalar from the field , so any
3:44real number , e.g. , 5 . And then you grab
3:48a vector , e.g. , 1 2 3 . And in between ,
3:51you put the multiplication with a
3:52circle . The multiplication is also
3:54defined by default as you already know
3:56it . Every component times the number .
3:59Of course , with a circle around it
4:01again , to distinguish it . As always ,
4:03for all this stuff to be a K-vector
4:05space , it needs a few conditions again .
4:07The first condition is actually four
4:09conditions at once . Ugh . The set V must
4:12form an abelian group with the plus , so
4:14it definitely has to be closed . In
4:17addition , there are the four other
4:18conditions for an abelian group . First ,
4:21the plus must be associative . So you
4:23can place brackets however you want
4:25when adding vectors . With our
4:27definition of vector plus , this is
4:29completely logical , just like in school
4:31. It doesn't matter where you place
4:33your brackets . You get the same result
4:35on both sides . Then , there should be a
4:37neutral element for the addition . So , a
4:40vector + e gives the vector again , and
4:42vice-versa . Why it's called e is
4:45obvious . In our example , that's the
4:47zero vector . We can add it to any
4:50vector and the result is the same
4:52vector . Third , every vector also has an
4:54inverse with respect to the plus . Added
4:57together , this should yield the neutral
4:59part . Again , just as you did in school ,
5:02adding the negative of the vector gives
5:040 0 . And we already said that is our
5:08neutral element e . And for it to be an
5:11abelian group , commutativity . So the
5:13order must not matter . V1 + V2 = V2 +
5:17V1 . Obviously . Order doesn't matter for
5:20vectors from school either . With that ,
5:22we've ticked off the conditions for an
5:24abelian group . But it continues . Number
5:275 . You should be able to distribute the
5:29multiplication with the circle and the
5:31addition with the circle . So , a scalar
5:33from the field times the sum of two
5:35vectors should be the same as the
5:37scalar times the first vector plus the
5:39scalar times the second vector . Let's
5:42see if it holds true for our example . A
5:44number times a sum of two vectors can
5:46be written down just like we're used to
5:48. We can then essentially multiply it
5:51out inside the vector and pull it apart
5:53again . Great . That works too . Likewise ,
5:56you should also be able to distribute
5:58the number-addition space . So , two
6:01numbers added together and multiplied
6:02by the vector should also be
6:04distributable . Sure , again with the
6:07example 5 + 6 multiplied by the vector
6:091 2 3 . First , we substitute again .
6:12Multiply it out there and pull it apart
6:15again . So that works too . The last two
6:19parts . The same as with the number
6:21addition should now also apply to
6:22number multiplication . Either you
6:24multiply both numbers first and put
6:26that in front of the vector , or you
6:28slap one number in front of the vector
6:30first and then the other . Does that
6:32work in our example ? First , substitute
6:35again . The order in which I multiply
6:37numbers doesn't matter . And if we
6:40rewrite that , we see it applies here
6:42too . Finally , the last part , the
6:45neutrality of the neutral element in
6:47the field . If you multiply your vector
6:50by the neutral element of the field ,
6:51the vector should simply come out again
6:53. And you all know that this applies to
6:56our example . One times a vector is
6:58equal to the vector . Wow , shoot , man ,
7:00that's quite a lot . If the examples
7:02went too fast , just hit pause . They are
7:05quite simple ; you'll understand them if
7:06you just look at them for a second . But
7:08so you can test if you've understood it
7:10, here is a video where we test whether
7:12something is a vector space . And
7:14because this whole thing was such a
7:15stupidly long definition by the math
7:17geeks , we'll keep it short today . Bye .