 Welcome back to our lecture series linear algebra done openly. As usual, I'm your professor today, Dr. Andrew Missildine. This will be the first video for section 1.3 in the textbook entitled Vector Spaces and just as a reminder, our goal in chapter 1 with regard to learning about linear algebra is really just to expose us to the primary ideas that linear algebra is all about. You can think of this as the who's who's of linear algebra or another way I like to think about it is like to under I'm trying to help us understand. What does the word linear actually mean now? We've learned in previous sections of of the book here about fields and linear equations So we learned a lot about what a linear equation meant like in previously talked about systems of linear equations the next important linear structure that we're going to discuss is the idea of a linear combination and This is a word that makes sense in the context of a vector space Now in order to define what a linear combination is we need to first understand what a vector is Now some of us might have some exposure to the idea of vectors in the past Perhaps like in a physics or other science or engineering course to a mathematician. These are Vectors certainly, but a vector can be much more than that Honestly when it comes to vectors, I often think about The villain from despicable me right who who's also named vector, right? He's committing crimes with both magnitude and direction and again That's often the physical interpretation of vectors But it turns out that in a mathematical setting a vector is going to be anything that we can add together or that we can scale that is we should be able to add together vectors and we should be able to Scale a vector by a different type of number, which we call scalars scalars We talked about already in the context of a vector field Vectors themselves are going to be different types of numbers which again borrowing the physics example a vector is a Is a mathematical quantity with both direction and magnitude So to formally define what a vector is we're going to first define the notion of a vector space a vector space As the name suggests is going to be a collection of vectors that satisfy certain conditions a vector space over a field Now our primary goal will be to discuss vector spaces over the real field the field of real numbers But honestly speaking the algebra we developed can be done over Various different fields such as the rational field the complex field of finite fields like Zp Just as an example right there are many more fields than just these four But these are the ones we're going to focus on in this class here So a vector space over a field is a non-empty set of which will typically call that v for vector space And the elements of that vector space we call vectors So if we were to stop right there This gives us a definition of what a vector is a vector is just an element of a vector space Well, what's a vector space? The definition is going to feel very similar to a field although the elements of a field remember we call scalars a vector space We'll have something which we call addition now because the field itself has addition Sometimes we have to distinguish these two additions which are called which are playing together here So for a vector space, this is sometimes called vector addition and the addition we saw over the field We might call scalar addition if we need to distinguish between the two So one operation will be vector addition and then the other operation will be scalar multiplication now unlike the Op the multiplication we defined for a field So over a field multiplication was you take two scalars and you multiply them together. You get a scalar in the context of a vector space scalar multiplication here means that we're going to take a Scalar which comes from our field F and we're going to combine it with a Vector which comes from the vector space and this produces a new vector All right now a convention I should mention that that we use when we describe vectors here is that scalars will generally be written as lower case alphabetic symbols like C and D or A and B oftentimes at the beginning of the alphabet we typically use letters at the end of the alphabet as for variables and Letters at the beginning of the alphabet for constants or unspecified arbitrary numbers And so that's how we'll typically denote a scalar on the other hand vectors We're typically going to use a bold face font so that when vectors and scale scalars Co-mingle with each other the vectors will be bolded the scalars will not have that bold face font And so it becomes easy for the reader to distinguish between vectors and scalars For so for example in this line right here You can see that these the you and V are bolded thus the represent vector quantities and then the scalars seed is not bolded Now when one writes on hand by hand bold face font is very difficult And so oftentimes you'll see that someone's right to vector like you and V But then we'll draw some type of little arrow over it to indicate that this is a vector quantity You'll see this notation commonly used throughout here Just wanted to mention that notation before we go on so a vector space is a set of objects Which are called vectors which has a vector addition and a scalar multiplication and just like a field there are certain axioms required For these two operations in order to call it a vector space now the list for vector space is not as long as it was for a field Although there are some things that are very similar so with describing Vector addition we require that vector addition be commutative U plus V is the same thing as V plus U. We require that vector addition Be associative that is you could redo parentheses however you want U plus V then adding W to it is the same thing as V plus W then adding U to it You can either add the first two first or the last two first and we'll make a difference We require there is some type of additive identity Associate to vector addition we get U plus what we call the zero vector U plus U Or sorry U plus zero is the same thing as zero plus you which is just you itself So an additive inverse and we also we want additive identities We also want additive inverses which just like a field will refer to the inverse of a vector U as negative U And this will have the property that U plus its inverse will just equal the zero vector so when you look at the axioms of the Edition over the scalars of on the field versus this vector addition. These are the exact four same properties We want commutivity associativity identities and inverses for a vector addition that parts the same What's going to be different of the axioms associated to scale and multiplication we're going to have some distributive laws So this one right here says if you add together two vectors, and then you scale it By a scalar you could actually distribute the scalar across the vector addition And you could get the scale that the scalar product of C and U with C and V and add that together That would be the same thing so we get we get a distribute distributor property across The addition here so scale and multiplication distributes over vector addition on the other hand the second So previously we might refer to this as the left distributive property Axiom six right here looks like the right distributive property. We can distribute a vector over this time This is scalar addition. This is what you have to be aware of here if you add together two scalars and then Multiply that by a vector. That's the same thing as scaling the vector and then adding them together And so this is this will can get sometimes confusing when one thinks about these abstract algebra notions We're using the symbol plus To mean two different things right here on the left-hand side that plus symbol means scalar addition We're adding together two scalar numbers and then we're multiplying that by a vector in contrast The plus sign on the right means vector addition We're adding together two vectors which is a different meaning of addition And we can distinguish between scalar addition and vector addition by context and this can be a little bit confusing at first But we get used to it the more practice we get Axiom seven here This is sometimes called homogeneity or to me this kind of looks like an associative property when it comes to scale and multiplication If you scale a vector by D and then you scale it by C This is the same thing as just scaling the vector by CD Which again you can redo the parentheses with regard to scalar multiplication And then lastly we want that scaling multiplication has an identity That is if you multiply by the number one which lives inside of our field F Then that's the same thing as just the vector itself scaling by one doesn't do anything Now we have a shorter list of axioms when it comes to a vector space And that's because the field of scalars already has a long list of axioms built into it And so a lot of nice properties can be Guaranteed because the field of scalars itself has that list of ten axioms for example We can take it for granted that our field has the number one in it because that's an axiom of the fields of the field of scalars there now I want to present you an example of a Vector space before we before we end this video right here now the example that many of us are probably already familiar with is the one That comes from from physics Where in that context vectors a Physical vector is just an arrow And our vector villain from despicable me This is the type of vector he was referencing when he gave his name there right a vector is a quantity with both magnitude and direction So we often denote this as a picture of an arrow in the plane or in three space in which case There's some type of direction associated to it right if we take the positive x-axis We could measure the angle our vector forms with the positive x-axis that gives us a direction of some kind We can also measure the length of the vector. So it has its length. How long is it? And that's what we often refer to as this magnitude The longer the vector the more Magnanimous it would be is that the word I should use here the more powerful the vector is like a longer force vector means It's a stronger force and so This is this is going to be like a numerical quantity because the magnitude itself will be a scalar quantity but and then The direction itself is also a scalar quantity But then the vector the arrow puts together two scalar quantities and force and creates this new object Which we call a vector In which case a two-dimensional vector would just be a magnitude with a single direction and three dimensions are higher We need multiple directions not just one to describe where it's pointing in space here Now if these are vectors we have to be able to add them together and we have to be able to scale them So what does it mean to first of all add vectors together? Well adding arrows. I mean what does that mean? I mean like what is what is Volkswagen plus Pikachu that equals happy day, you know What does it mean to add things which are not real numbers anymore, right? We have to in some regard define what addition means in this context. So when you have two arrows So take these two right here, for example, there's the arrow u and there's the arrow v Now one thing you should be aware of when it comes to these physical vectors like so if you take force vectors For example, the location of the arrow in space doesn't matter, right? It's only the direction in the magnitude that matters So if you draw the same arrow pointing the same direction the same length all three of these vectors Should be considered the same quantity from a physics point of view and as such we can position our vector so that they have a common point Like each each of these arrows here It has the head which is where the arrow is pointing and then it has its tail If you're in archery, this would be where your your feathers are, right? We could position the arrow so that they have a common origin common The tails touch each other so position the tails and then form a parallelogram using those two vectors What I mean is you're going to take the vector One of the vectors you and you're going to copy it and put that copy at the head of v over here And then copy the vector v and put it at the tip put its tail at the head of you And this will always form a parallelogram in the plane Now even if you have three dimensional vectors or higher dimensional vectors if you take two vectors The their common tail plus their two different heads will give you three points pretend most likely non Collinear points and these three points will always determine a unique plane and whatever dimension you're playing around in So this this makes sense to focus just on two-dimensional pictures right here These two vectors together will always find form a parallelogram And if you connect the dots from one corner of the parallelogram to the other corner of the parallelogram That vector is will be defined to be u plus v. That's the sum of the two Vectors and because of this geometric principle adding together Vectors physical vectors is often referred to as the parallelogram rule we take the diagonal of The parallelogram associated to these two vectors or another way of thinking about it Is that instead of putting the two vectors so they share common tail you could put that you have one vector and then at the head of the of the first vector you add the second vector like so and then The sum would be the net vector that goes from the tail of the first one to the head of the second one This approach can be very advantageous when we start adding together multiple vectors Because you can have a vector going here and then a vector going here and then a vector going here and then a vector going here In which case the sum of the vectors the net vector Will just be the head will go from the tail of the first vector to the head of the last vector So this would be the sum of all of those vectors. It's a very nice geometric principle known as the parallelogram rule That's how one can add together vectors. How do you once scale vectors? Well, there's two things to consider if you have a vector and you times it by negative one You want to create the additive inverse what you do is you just take the exact same the same vector with the exact same Length you're just going to switch the direction right so you just put you to switch the the head with the tail That's all you have to do to form the inverse vector And so now it'll be pointing in the opposite direction The reason why we we want this to be our inverse vector Is that if you put these things head to tail right by the parallelogram rule we learned a moment ago If you go this way and then you go back this way you're going to end up with just a single point The net effect is just you go a single point and this is what we mean by the zero vector in this in this context The zero vector is the vector with no magnitude It doesn't move whatsoever And so the way to cancel out a vector will just be to take that same vector and point in the other direction with the same magnitude That's what it means to multiply a vector by negative one What it doesn't mean to multiply a vector by some positive real number Well, if you have a vector v like you have right here Scaling it means to elongate the vector by that quantity. So you see illustrated here three v It'll be the same direction as the original vector, but now it's going to be three times longer the magnitude got multiplied by Uh that amount right and so you could so multiply a vector by a positive number larger than one would make it get longer By a small number like between zero and one would make it get shorter And so we can physically dilate or contract vectors using scalar multiplication And so this is how one defines for physical vectors The idea is a vector addition and scalar multiplication and one could check that these definitions of addition and scalar multiplication Satisfy the eight axioms of the vector space So these physical notions of vectors these arrows in fact form a vector space And this is one of the most canonical examples of a vector space