What is a Vector Space? (Abstract Algebra)

you probably first encountered vectors in physics where they are commonly used to describe forces in science a vector is an arrow with a length and a direction you can have two-dimensional vectors which live in the plane or three-dimensional vectors which are more appropriate for the three-dimensional space that surrounds us taken together we call the collection of vectors effectors space but you can generalize this idea and find countless other uses for vector spaces in the plane of vectors described by pair of numbers the x coordinate and y coordinate it's assumed that the tail of the vector is at the origin while the head of the vector is at the given coordinates we call all the vectors in a plane a 2-dimensional vector space because it takes two numbers to describe each vector you can add any two vectors to get a third vector and if you multiply the coordinates by a number you can scale a vector similarly you can have vectors with three coordinates and x y and z coordinate once again the tail of the vector is at the origin and the head is at the point given by the three coordinates like before if you add two vectors in three-dimensional space you'll get a third vector and again you can scale vectors by multiplying the coordinates by any real number all of these vectors form a three-dimensional vector space because it takes three numbers to describe each vector it's tempting to think that two and three-dimensional vector spaces are all that we really need perhaps the fourth dimension if you work in space-time but this is abstract algebra and our goal is to generalize things as much as possible without sacrificing usefulness for example imagine an alternate universe with ten dimensions taking inspiration from two and three dimensional vector spaces we could suggest that in this universe forces could be described by a list of ten numbers and like before you can add and scale vectors we would call this ten dimensional space if you were to generalize this further you could say n-dimensional space is a place where forces are described by vectors with end coordinates we call this n space let's stop for a moment and look at what properties all of these vector spaces have in common for starters you can add any two vectors so there's an operation addition better still the order in which you add vectors does not matter vector addition is commutative there is a zero vector which doesn't point in any direction adding the zero vector to any vector leaves it unchanged so the zero vector is an identity element also for each vector there is another vector which points in the opposite direction when you add these two together you get the zero vector vector addition is also associative taken together we see that vectors form a commutative group under addition but vector spaces are more than commutative groups you can scale vectors by multiplying each coordinate by a real number when you do this the number is called a scalar these scalars give vector spaces additional structure and features beyond ordinary groups and for the record there are few properties of scalars which will probably not surprise you but they need to be said multiplying by scalars follows the distributive properties scalar multiplication is associative and multiplying a vector by one leaves it unchanged we're almost ready to give the textbook definition of a vector space up until now we've thought of vectors as arrows in space first in two and three dimensions then in any number of dimensions we saw that in all of these cases vectors were a commutative group with real numbers scalars what if we took this as our definition of a vector space would it cover more ground than just describing forces in space the answer is a resounding yes if we step back from this idea that vectors have to have a physical meaning we get something much more abstract and much more powerful and to take things even further what if we threw out the idea that scalars have to be real numbers and instead said they could come from any field like the complex numbers now we have a vector space let's take a breath and go over the abstract algebra definition of a vector space first you have a commutative group and the elements in this group are called vectors next there's a field of scalars you can multiply any vector in the group by any scalar in the field and you'll get a new vector the operation between scalars and vectors obeys the distributive and associative properties and lastly vectors are unchanged when multiplied by one here we have the full definition of a vector space as mentioned earlier this more general definition encompasses many more situations let's see if you for example consider all 2x3 real matrices these matrices form a commutative group you have zero matrix inverses and you can even multiply any matrix by a real number to get a scaled matrix the two by three real matrices form a vector space and it's a good exercise to check the remaining properties for another example consider all polynomials with real coefficients of degree 5 or less this set also forms a vector space if you add two polynomials of degree 5 or less you'll get another polynomial of degree 5 or less and notice how I keep saying or less this it's key look what happens if you add these two polynomials of degree 5 you get a polynomial of degree 3 so when you add two polynomials in this group the degree may go down but it will never go up and once again you can multiply any polynomial by a real number and get another polynomial of the same degree unless you multiply by zero let's see one more example consider the set of functions which are continuous over all real numbers in calculus you learn that if you add two continuous functions you get another continuous function it's also good practice to check that multiply and continuous function by a real number doesn't cause any problems it's still a continuous function this is a great example of a vector space that really departs from the original idea of arrows in space vectors can be pointy things Kawa I'm not sure what these vectors are pointing at

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Keywords: Socratica, SocraticaMath, abstract algebra, vector space, vector spaces, scalar, scalars, math, maths, educational video, tutor, college, university, advanced math, higher math, modern algebra, STEM

Id: ozwodzD5bJM

Length: 6min 57sec (417 seconds)

Published: Thu Oct 20 2016