Birne Binegar: Okay welcome back to math 3013 linear algebra.
Birne Binegar: Okay, so um I guess I just started asking you, if there any questions out there.
Birne Binegar: Okay hearing none let's start with with reviewing what transpired on Monday and then we'll make a little bit of more progress towards.
Birne Binegar: Understanding the basic objects of this course, which are vectors inside an n dimensional vector space.
Birne Binegar: Okay, so.
Birne Binegar: Last time I introduce now to really basic items.
Birne Binegar: But they're similar to one another, so I want to clarify again a distinction between them now there's a distinction between sets and ordered lists, we talked about a set we're just talking about a collection off which we not putting it in any particular order.
Birne Binegar: To think through them as a group.
Birne Binegar: We use curly braces typically you know element to the set so here, I have written down.
Birne Binegar: Set contains three elements and each elements happens to be a number one minus three inch were to.
spot.
Birne Binegar: example the set an ordered list it's similar to this in fact you can always get a set from an ordered list by simply ignoring the ordering of elements but.
Birne Binegar: In an order list, though the ordering is really crucial and that's why we make the distinction between ordered list and set, we also make a distinction location wise as well, using either.
parentheses to include.
Birne Binegar: This or we use square braces and depending on whether we use parentheses or wear braces depends on the context in which we use what we'll see is will typically use square braces when we're trying to do, new vectors as ordered list Okay, so, in fact.
Birne Binegar: One of the things we do find a Monday was what a vector is inside this linear algebra course and.
Birne Binegar: What it is ordered list of real numbers.
Birne Binegar: write them down my taking a set of numbers.
Birne Binegar: Order and then forming a list from.
Colin Myers: them.
Birne Binegar: Right boom an order in clarity closing them and square braces.
Birne Binegar: Right and then the number of numbers that go into the list is the dimension of the vector and the individual numbers that appear in the list are called a component of the vector.
Birne Binegar: So in this example Bob we have a three dimensional vector which component minus ones five in the square root to.
Birne Binegar: All right, and then, because these are order plus.
Birne Binegar: Two vectors are not going to be equal unless the numbers that uh, not only did the numbers have to be the same in each word list, but they have to be written in exactly the same order in order to be regarded as equal.
Birne Binegar: All right now The other thing i'm going to review real quickly is two ways of specifying set one is as we've done before we simply.
Birne Binegar: write down the element in some order, but the order is immaterial you write them down in some order and we see closing curly braces and that's that's defined a set very concretely.
Birne Binegar: But an alternative way and actually much more common in this course is to specify your set by stating a property or characteristic that distinguishes the young to that set so, for example, i'm.
Birne Binegar: Not set so i've written the Bob.
Birne Binegar: could also be have been written at the set of real numbers, such that their square is equal to two.
Birne Binegar: Alright, so.
Birne Binegar: What i've done is i've said verbally what this notation below indicates.
Birne Binegar: when constructing a set so that's why i'm using the curly braces the basic objects, are going to be real real numbers.
Birne Binegar: But we're going to put a condition on the real numbers that's The purpose of this vertical line in the Center the set you read that as such that so s is the set of real numbers or x's that are real numbers, such that they're squared is equal to two.
Birne Binegar: yeah i'm also introducing a.
Birne Binegar: This triple line, this seems to be stronger than an equal and that I use this to indicate a mathematical definition so so it's actually defined by.
Birne Binegar: The expression on the right hand side.
Birne Binegar: And then the symbol, now that vertical line years you read that is such that so so is the set of.
Birne Binegar: elements.
Birne Binegar: accident happened to be.
Birne Binegar: Real numbers, such that they're squares equal to one.
Birne Binegar: All right, and notice, so these two sets are the same, whether you find it by explicitly listening the element already set up the set by saying what I make it from in the condition that the elements have to satisfy.
Birne Binegar: Okay don't actually what.
Birne Binegar: I want to point out here is that we can write sets differently until I see this as an equality, and we can compare element by element and verify that true but, more generally.
Birne Binegar: When we have we specify such using properties are conditioned it's not so easy to see that they're actually the same set.
Birne Binegar: So I just want to remind you that if you want to show that two sets are equal either you have to compare them directly element by element.
Birne Binegar: Or you have to demonstrate that anytime you have an element one set, it has to belong to the other set in simultaneously any element of the second set must.
Birne Binegar: also belong to the first step is that they both these conditions are true in only both these conditions are true are the two sets of equal, so another way of putting this two sets are equal, if each is, this is a subset of the other.
Birne Binegar: But know that there's two conditions to check.
Birne Binegar: to verify this single equality.
Birne Binegar: Okay um one last thing about such, and that is there Cartesian products.
Birne Binegar: So here, this is sort of a notation or device.
Birne Binegar: You have to set A and B if I write a craft beer eight times the, this is the set.
Birne Binegar: itself it's a set and it's consistent all order pairs A and B so here on still using.
Birne Binegar: i'm indicating ordered list using parentheses, rather than square brackets, but this is still an ordered list a an ordered list with two elements a proceeding be in that list.
Birne Binegar: Okay until.
Birne Binegar: More set they're already notation now the same definition would be written a times B is defined as this set of order pairs such that the first element is an A in the second element is always in the city.
Birne Binegar: Okay, and then we have the two fundamental sets that will use all the time and not linear algebra and they have their own to both we denote by this funny fun capital are the set of real numbers, so that fun exactly called.
Birne Binegar: blackboard bold got double line to everywhere possible yes and it denotes a set of real numbers, so not only the integers 12347 etc, the rational numbers 789 hundred 99 1000.
Birne Binegar: And the irrational numbers, like the pie, the square root to accept all the real numbers, now the numbers that are typically you're using calculus and all applications.
Birne Binegar: All right, and then distinct from headset is the set are super script and which is the set of all and dimensional vectors in here in the set theoretical location I indicated.
Birne Binegar: That vector space on an as a set consisting of ordered list of an element where each element in each list is actually a real number so it's a setup ordered list of in real numbers, just before.
Birne Binegar: And I point out that the ellipses here in this notation are simply the indicated pattern that which began continues until conclude with axiom.
Birne Binegar: And so now yeah so.
Birne Binegar: This notation verbally of saying that aren't in is a set of order list of in number in real numbers.
Birne Binegar: One thing I want to point out, though, is how this or notation for the set of n dimensional vectors mimics its construction using Cartesian products.
Birne Binegar: So you see that are effectively squared is the Cartesian product of our of itself and, similarly, are the third could be regarded as the Cartesian product of our itself three times.
Birne Binegar: Okay, but remember this is not multiplication of integers or multiplication of real numbers, this is this Cartesian product is taking an element from the first set in an element from the second set and combining it into an ordered list.
Birne Binegar: Okay, so.
let's see.
Birne Binegar: Is there any amount of order lists.
Birne Binegar: So the number in is indicating how long the list is going to be so.
Birne Binegar: For an n dimensional vector you want to list containing exactly in numbers.
Birne Binegar: Okay, so, then this is the capital N is indicating the size of the list, how long it is.
Birne Binegar: But we use that.
Birne Binegar: to distinguish between one dimensional vectors which basically all have to live on line two dimensional vectors which could stand playing.
Birne Binegar: Three dimensional vectors would could generate a three dimensional space, so this and it's not you'll you'll think of it primarily is how many components of vector has, but it also indicates the dimension of the space in which these vectors live.
Birne Binegar: Okay.
Birne Binegar: So let's go back to linear algebra and or basic vector space are super script in.
Birne Binegar: So what i'm going to do next is i'm going to introduce the two fundamental mathematical operations that will perform all the time in working with n dimensional vectors.
Birne Binegar: will learn how to add them and we'll learn how to multiply vectors by numbers now here i'm using the Cartesian product location, so what this vector edition is it's up.
Birne Binegar: This plus is just a symbol used to indicate the operation and then it's taking a pair of that ders see this Cartesian product means i'm going to take.
Birne Binegar: One element of our and i'm going to take another element of our and form a list and then map that to another n dimensional vector.
Birne Binegar: On scale of multiplication what we do is, we take a number simply a real number and an n dimensional vector somehow I combine them in produce another n dimensional vector.
Birne Binegar: Okay, so this is all happening, occasionally, but I want you to get used to seeing this location and realizing what it means mathematically because it's not clear explicitly from the location, that if I had.
Birne Binegar: A vector say D one, two and three, and I had a vector w let me put little arrows over those that maybe 101 that.
Birne Binegar: No well what i'm trying to describe is how i'm going to add.
Birne Binegar: These two vendors.
Birne Binegar: Right so.
Birne Binegar: The vector edition it's going to take a pair of vectors and that's what i'm indicating with this Cartesian product location.
Birne Binegar: But let me, let me just proceed forward and being more much more explicit suppose we have two n dimensional vectors X and y.
Birne Binegar: O uses shorthand notation where I use a blackboard bold, to actually indicate all in component of a single n dimensional vector so that vector has an component, but it's a single vector in itself in Why is a vector that's distinct from X at least know occasionally.
Birne Binegar: We add that here's here's the rule we take the first component, the first factor.
Birne Binegar: add it to the first component of the.
Birne Binegar: Second factor, and that will be the first component of the sun axes.
Birne Binegar: And the second component is of the sum is created by adding the second component of the individual summoned alright, so you should be familiar with this operation, either from freshman physics or calculus one, so we we add vectors in the usual way component by component.
Birne Binegar: and
Birne Binegar: here's an explosive example.
Birne Binegar: Look more complicated is one plus two is equal to three three plus zero is equal to three minus one plus one is equal to zero.
Birne Binegar: The second really fundamental basic operation is scale and multiplication and hear what you do is you start with a number and you a vector and you combine them as follows.
Birne Binegar: So on the left hand side you can think of this as a symbol that's going to represent the scale or multiple of the vector X by the number of Lambda in on the right hand side, I give you a formula for that.
Birne Binegar: it's going to be, obviously in n dimensional vector because you see n components here and the components of this scale or product or the components of the original vector each multiply by the same number Lambda.
Birne Binegar: Alright, so here's the sample computation if I.
Birne Binegar: Take the vector 132 and I multiplied by the number minus two, the result is the vector minus two minus six minus four.
Birne Binegar: Okay now this stuff is all really rudimentary and you should be bored to tears with this, but.
Birne Binegar: The real point of this is that these this set aren't in in these two basic operations are the basis of the whole course.
Birne Binegar: And what will get derived from these basic definition is a lot of additional structure structure that can be used to analyze much more complicated things and you would think that you could describe it by just ordered list of numbers.
Birne Binegar: But it's going to get abstract, so the emphasis right now is how concrete the basic definitions are.
Birne Binegar: All right, so.
Birne Binegar: there'll be another point to them i'm just about to make.
Birne Binegar: What i've described up to this point in today's lecture is vectors as.
Birne Binegar: ordered listening numbers, but we have other ways about a thinking of vectors we can think about them visually as being represented by line segments, with a particular direction in a in a certain length as well.
Birne Binegar: So that raises the question how do I pass between the two different renditions of what a vector should be is it a directed line segment, or is it an ordered list of numbers well in some sense it's both.
Birne Binegar: What we have to do is figure out how to correlate these two different ways of looking at vectors and so to correlate.
Birne Binegar: A two dimensional vector vector with two components to.
Birne Binegar: You can relate this to a directed line segments it's drawn and a two dimensional plane.
Birne Binegar: In the way you do, that is, you well let's go from directed line segment to order list you start with a directed line segment you establish.
Birne Binegar: A coordinate system whose origins at the base of your directed line segment and then you use a ruler, to mark out the coordinate grid.
Birne Binegar: And then you identify the coordinates of the tip of the vector relative to the origin that placed at the base of the vector and then that gives you an order list of numbers that you'll say if the let's call it the linear algebra a vector that corresponds to this geometric vector.
Birne Binegar: Alright, so that's how you would pass from an intuitive idea of what a vector is to an actual calculation will vector order list of numbers that are really just designed for calculations.
Birne Binegar: When you can go the opposite way as well now here what you do if you have an ordered list of numbers, you created coordinate grid and then.
Birne Binegar: You draw a line that starts at the origin of that coordinate grid to the coordinates that are specified by your ordered list.
Birne Binegar: And then you can erase the coordinate grid and you're left with a directed line second so you can go back and forth between these two pictures of what a vector is you as an ordered list of numbers in this geometric rendition of the vector.
Birne Binegar: So this actually brings up a very important point in this course, and now, that is the fact that somehow there's an abstract thing the vector and there's different ways of.
Birne Binegar: Dealing with it, you can deal with it that's a geometric object, or you can deal with it as an ordered list of numbers but somehow there's an abstract thing that.
Birne Binegar: encapsulates both this is sort of like the number three representing lots of things, it could represent three pens, it could represent three fingers, there are many different versions of the number three and.
Birne Binegar: Even symbol that's a similar way, you should think about vectors there's an abstract idea, like the number three and there's various renditions of that idea.
Birne Binegar: Okay, so we have a rule for adding vectors when we regard them as ordered list of numbers there's a similar rule that we can use directly on vectors thought of as direct as line segments.
Birne Binegar: So now here's how that goes, I suppose I want to add these two directors behind segment well the rule for doing so issue.
Birne Binegar: Now, remember, for these directed line segments their position on the on the piece of paper is in material, you can move them around, but the one thing you can't change is the length of the vector or its direction.
Birne Binegar: So we call that parallel transport where we move a vector around on a piece of paper without changing its length or its direction.
Birne Binegar: So what I can do to add these two directed line segment is I can move the second one, so that is base is that the tip just sort of slide it over.
Birne Binegar: And then I can draw a line that starts from the base of the first factor to the tip of the second factor, and then I can erase the original to summons up here and i'm left with a directed line segment that's going to be the result of adding the first directed line segment second draft.
Birne Binegar: And in fact.
Birne Binegar: This geometric method of adding vectors is completely compatible with our method of adding vectors is ordered list of numbers by convert this to an ordered list of numbers to work, the second victor.
Birne Binegar: list to an order alicia numbers add those together and then convert back to a geometric vector it would replicate the vector right now i've got here at the bottom of the screen done using purely geometric operations.
Birne Binegar: Okay, so.
Birne Binegar: scaling multiplication while even easier than vector addition when we think of vectors as ordered list of numbers it's actually a little bit more complicated when we think about a geometrically.
Birne Binegar: Basically what it does, is it changes the length of the vector, however.
Birne Binegar: It does more than that, if the scale or multiple number Lambda that you're going to multiply vector by if that happens to be a negative number, it not only changes the link it actually reverses the direction it doesn't change it to an arbitrary direction it exactly reverses the direction.
Birne Binegar: So let me show you how this works by.
Birne Binegar: Looking at both the order list of numbers in a geometric vector So if I had an ordered list to one and I scale and multiply it by the number of minus two, I get the vector minus 42.
Birne Binegar: So let's see what happens.
Birne Binegar: So linear algebra basically Gala multiplication of the vector to one by the number minus two produces this the ordered list minus four minus do but geometrically what happens is this vector is directed line segment.
Birne Binegar: Whose tip is that the coordinate point to want this gets sent to a different directed line segment and what I hope you see in this picture is this direct line segment.
Birne Binegar: Is it's just.
Birne Binegar: Essentially, hate it heading in the same direction as the original vector but because of the multiply by negative number it's actually heading in the opposite direction and it's linked it's also changed it's changed by a factor of two.
Birne Binegar: Because the magnitude of the scale of factors to.
Birne Binegar: Okay, so in this course i'm going to emphasize always vectors as ordered list of numbers.
Birne Binegar: But to communicate and pass on understanding of what's going on sort of an abstract or that intuitive level often resort to a geometric picture, so you have an alternative way of looking things and hopefully i'll turn the other way of confirming result.
Birne Binegar: Okay, and more importantly, the reason why we work with order list of numbers is because I can really only draw geometric vectors on a piece of paper, I can detect them if I have good 3D.
Birne Binegar: Know graphic software, I can maybe pick them on a computer screen via some three dimensional picture of you have Dr maybe an explicit.
Birne Binegar: Three dimensional picture, but if i'm beyond three dimensions, I really can't even.
Birne Binegar: visualize a vector as a directed line segment because I don't have any any arena that's four dimensional that I can actually visualize Although some people claim, and if you include time you can visualize it but.
Birne Binegar: First of all, purpose, you cannot put it down on a piece of paper and see it as concretely as simply a direct in line segment on a piece of paper so.
Birne Binegar: we're dealing with more general vectors vectors living in a four dimensional space and beyond, really, our only resource is to treat vectors as ordered list of numbers and then essentially all vectors.
Birne Binegar: behave, the same as ordered list of numbers.
Birne Binegar: Okay, so now.
Birne Binegar: If I were marking the first lecture you know no data coming through this course with an idea of vectors as as they occur in physics and engineering where they represent physical quantities of direction magnitude.
Birne Binegar: and your physic courses you'll often see when in figures and pictures and diagrams of what's going on vectors represented as directed line segment.
Birne Binegar: that's um.
Birne Binegar: that's.
Birne Binegar: Hopefully familiar to you, but in fact there are other objects that occur in physics and engineering that behave like that jurors but only when you view them abstractly enough you see the vector like what qualities, so let me give you one example of that.
Birne Binegar: In acoustics.
Birne Binegar: Air pressure fluctuations very sound waves can be represented this factors, now that should a poli right about here and annoy you how do I see its direction, how do I see it, magnitude, how do I see its components as or list of numbers, well, in fact.
Birne Binegar: Neither of those renditions of what a vector is is going to help you understand how to think of air pressure fluctuations as a vector rather.
Birne Binegar: The way, you will understand them, your pressure.
Birne Binegar: fluctuations are sound waves as being vectors is to understand the operations that you can perform on them and how those operations make.
Birne Binegar: Sound waves up behave like vectors So what is adding vectors mean when I think of the vectors as sound waves well, you can combine sounds.
Birne Binegar: If you listen to an orchestra, you can know you hear the total Orchestra, the sound of the violin section, the sound of the trumpet section and the sound of the individual violins and trumpets.
Birne Binegar: So now, the way we would think of US sound way as a vector well is because we can add them together, simply by combining sounds as you would and they will simply combined you, you are here multiple sounds rather than single sound if you're listening to.
Birne Binegar: an orchestra okay so and sometimes we have a way of adding.
Birne Binegar: Sound waves.
Birne Binegar: Now, what about scale and multiplication well scale an application that can be implemented by increasing the volume or decreasing the volume of a sand or what we call the amplitude of the vibration.
Birne Binegar: And we can change that by a positive number we just re skill, a vibration, making it half the amplitude or twice the entity and we can modify in it soundwave that way and so that would be a legitimate operation, we could perform on sound waves.
Birne Binegar: But what about multiplication by a negative number well what that does.
Birne Binegar: Is you really have to visualize how the lesson for soundwave how the pressure fluctuate so if this is a sound wave in here you think of time going in this direction and.
Birne Binegar: Not sound wave the air pressure just fluctuating this site new solo fashion well what multiplication by a net negative number like minus one would do is it would.
Birne Binegar: Think of this as the equilibrium point where there's epic no pressure here we have large negative pressure large positive pressure.
Birne Binegar: One regardless multiplication by minus one we've converted this sound way into one worthy and worthy yeah well I guess displacement from the equilibrium prison is reversed, so at this point here.
Birne Binegar: I have a negative.
Birne Binegar: Negative displacement I would convert that to a positive displacement, so you get something like that, so this multiplication by nine is minus one without changing amplitude of the of the oscillation, but it would chase now, it will change by 180 degrees, the phase of the oscillation.
Birne Binegar: Okay.
Birne Binegar: Oh, I have a question here when adding two vectors do both sets have to have the same length and yes that's true you cannot add a two dimensional vector to a three dimensional vector because there'll be one strangling component, it has no rule.
Birne Binegar: To.
Birne Binegar: To compute with.
Birne Binegar: Okay, so yeah I just lined up with the chat and try to answer a question that appeared there.
Birne Binegar: Okay, so um.
Birne Binegar: So in fact I should remark that, even though this seems like an extraneous operation and i'm sort of forcing some notion of.
Birne Binegar: That tradition and scale of multiplication on oscillations, but in fact this is very practical now when you change the face of oscillation by multiplying it sound wave by minus one, this is actually the basic principle by without noise cancelling headphones work.
Birne Binegar: Inside your headphone you have a small microphone that captures the outside, noise, then you have an electronic component that reverses the face of that oscillation.
Birne Binegar: And then inside the headphone you have little speaker and what it does, is it tries to replicate the exact opposite of the external noise.
Birne Binegar: And it has that to the sound coming out of these not signal going into your headphones so that the external noises cancel, leaving only the internal lose noise that's coming from the signal into the headphones.
Birne Binegar: So, in fact, these three vector operate these two vector operations have a very practical application in acoustics and moreover they show you.
Birne Binegar: What we'll see later on, in fact, it gives this instead of all possible sound waves, the structure of what we regard as a vector space.
Birne Binegar: And the formal theory for manipulating three dimensional vectors or four dimensional vectors that's all going to carry over and allow us to manipulate sound waves in a similar fashion.
Birne Binegar: Okay.
Birne Binegar: So.
Birne Binegar: Now we're getting into.
Birne Binegar: Further development we have two operations director edition and and scalable application sorry this is this line is not written correctly what I meant to say the operations of scale or addition and.
Birne Binegar: vector edition and scale of multiplication that you can combine those operations so ignore the topic typo in the first sentence and.
Birne Binegar: realized that if I have a pair of numbers, and I have a pair of vectors, then I can combine all four of these objects and get a single vector.
Birne Binegar: Then, this would be the construction you take first number you multiply the first number get scalable vector a by the number of Lambda.
Birne Binegar: And then we can add to that the scale or multiple the vector be by the number new and the result of this combining these two operation to which is actually three.
Birne Binegar: You can have to scale them will.
Birne Binegar: play once scalable play again and then add to this actually represents three basic operations, but the end result is going to be a single vector.
Birne Binegar: We call constructions like this linear combinations More generally, we can have a set of end vectors instead of n numbers and we can create an even more general linear combination by adding together multiple scale of multiple multiple scale multiples of.
Birne Binegar: A list of investors now again because called a linear combination, and in fact this is this construction.
Birne Binegar: Making other vectors from a given list of vectors this is really, really common in this course.
Birne Binegar: Alright i'm also but less common but frequently useful in applications is the notion of vector subtraction i'm not introducing this as a fundamental operation like scalable application decoration because in fact scale multiplication can be understood as a particular linear combination.
Birne Binegar: A particular way of multiplying scale or multiplying vectors and then adding them together what this vector subtraction is it's the addition of one factor to the scale or multiple the second vector by the number minus one.
Birne Binegar: And what it's used for in applications is to indicate that displacement between two vectors so let me show you the.
Birne Binegar: geometric picture of what this vector subtraction does if you have two vectors A and B.
Birne Binegar: portrayed as directed line segment, there are different, a minus B is going to be the directors line segment, but which.
Birne Binegar: The base is at the tip of the first or the thing you're subtracting and the tip of the result is the same as the tip of the.
Birne Binegar: vector from which be is being subtracted.
Birne Binegar: Okay.
Birne Binegar: Oh, I have a question, this will this i'd be recording this lecture I should be recording this lecture now what happened on Mondays I thought I was recording and.
Birne Binegar: I pause the vector the recording when I started the lecture because I had started early and then I forgot to unpopular, so what I did on canvas is replaced the zoom lecture for the first lecture with the first lecture of my second.
Birne Binegar: 1130 section of linear algebra and how frequently do that if I flub up something in one lecture the video on will post will be the better copy.
Birne Binegar: Of the two lectures that I gave on that given day So yes, i'm recording the lectures on posting it on canvas but the one that actually posted is going to be the better version of the two lectures I I gave on on a given day.
Birne Binegar: Well they'll be any homework or quiz look like someone who were trying to avoid class.
Birne Binegar: know that when there is a homework assignment it's already posted on web design, but i'm not planning on giving you a quiz this week.
Birne Binegar: Okay, so.
Birne Binegar: let's move on now, the zero vector is.
Birne Binegar: Simply the vector from which every component is equal to 00 here's the zero vector inside the vector space are three.
Birne Binegar: Two or three is the order list of three real numbers and here's a particular order real lists of three numbers right 00 and zero that's called the zero vector and, just like the number zero and algebra.
Birne Binegar: If you add this particular vector to any other vector you get back what you started off with a is going, and moreover.
Birne Binegar: The way you can tell if two vectors are equal, is, if you subtract them one from the other you get the zero vector.
Birne Binegar: So two vectors are going to be the same if, and only if there are differences zero now this looks like an algebraic equation, I mean an algebraic equation in the Center of algebra from high school but.
Birne Binegar: What you don't understand this, is that a vector a is equal to a vector be if, and only if the difference between A and B is zero vector.
Birne Binegar: So what i'm going to caution you about is that what notation only when I write a vector a plus B by simply write two letters down a plus B you don't know what it is.
Birne Binegar: What is a plus B, well, it depends on what a B is it a B or numbers, then I would add these like numbers.
Birne Binegar: If i'm interpreting A and B is vectors I need to use the rule for adding vectors but occasionally it's going to look the same as if we're adding numbers and that's going to cause some confusion initially.
Birne Binegar: When you see some symbols on a page you're going to first have to recall what the symbol represents it's not going to be enough to simply see you're going to combine these two guys, because the rule for combining them depends on what the object that represents says.
Birne Binegar: So be wary of that and that's why it's very important whenever possible to actually decorate your symbol somehow to indicate what they represent, so my pretty little arrows over this.
Birne Binegar: Latin lowercase letter been talking, I want you to think of this, not as a simply the letter a I want you to think of this as representing a vector okay.
Birne Binegar: Oh i'm sorry you can't see the top is that better.
Birne Binegar: So the station, I was making if I simply write a plus B down you don't know what if you don't know what a vr as mathematical objects you can't really combine them.
Birne Binegar: But what i'll try to do to help you understand what's going on it's all frequently, at least on handwritten note adorn my vectors with little arrows of the top, so you don't think of these that if this is added to numbers but rather is adding two vectors.
Birne Binegar: Okay.
Birne Binegar: So um.
Birne Binegar: yeah inside yeah at the bottom of this page I just tell you about how i'm going to try to.
Birne Binegar: keep things clear by denoting vectors by putting little arrows over them in my lecture notes I don't have to resort to arrows.
Birne Binegar: i'm typically using a boldface fine to indicate vectors and then, if I want to indicate the number zero or i'll use the same.
Birne Binegar: symbol effectively zero, but you won't be written in boldface so on the left hand side, I have the vector zero in on the right hand side, I have the two components, each of which is the number zero.
Birne Binegar: Okay, so um that's as far as I expected to get today.
Birne Binegar: Let me just stop here and.
Birne Binegar: Now what we'll do i'm starting on Fridays lecture is i've got all the apparatus set up and now we'll start to get into what this linear algebra it's good for and.
Birne Binegar: know first thing we'll start to look at is solving systems of linear equations and relating them to matrices so on that I have one more fundamental object to introduce.
Birne Binegar: And that is the notion of what a matrix is but i'll introduce that not directly it's basically an ordered list of factors, but rather i'm going to introduce this is from an application that matrix these are going to arise as soon as you start to consider.
Birne Binegar: How the soul large systems of linear equations.
Birne Binegar: Okay don't put that off till Friday and i'll stick around in case there are additional questions up sorry, there is a question coming in, on the chat so the zero vector is a linear combination of any choice of vectors V1 to be in choosing all scales to be zero.
Birne Binegar: yeah yeah.
Birne Binegar: This is good i'm really happy, people are are are writing question doubt because it's the language that we need to get straight here so.
Birne Binegar: What what we're going to really kind of the essential and understanding material in this course is to go back to the very basic definitions, so a vector.
Birne Binegar: This is going to always be an ordered list of number.
Birne Binegar: Right and then the zero vector.
Birne Binegar: Well, in particular it's going to be a vector so it's going to be an ordered list of numbers but use particular numbers.
Thomas Kidd: And that's a paper.
Birne Binegar: i'm sorry.
Birne Binegar: That better for the zero vector now.
Birne Binegar: Talking about a particular vector and so a particular ordered list of numbers and the way you get the zero vector that's in three dimensional space is you simply write the number 03 times and enclose it in square braces.
Birne Binegar: Okay, so this is a particular example.
Birne Binegar: Of the abstract idea.
Birne Binegar: Then we're going to constantly be doing us will have abstract objects that are going to.
Birne Binegar: boil down to our basic definition and then we're going to have explicit renditions of these things, and these are going to be explicit concrete vectors notice that.
Birne Binegar: I say vectors in order list of numbers, I don't have anything to calculate with However, when I give you a particular example here i've got a list of numbers and i'll begin you least you're seeing numbers here, and that should hint that there's something you can calculate with now.
Birne Binegar: Okay, so that that i'm john from abstraction to concrete realization we're going to be doing this, all the time and it's important and questions like.
Birne Binegar: your initial one zero vector linear combination of choices that you see this is this is confusing you because you're.
Birne Binegar: you're grabbing things that occurred in various parts portion of the lecture but, really, the idea is to go back to the basics in constructing from the basis rather than sort of fuzzy ideas of the things that went on okay.
Birne Binegar: So we can move up the paper yeah well that's all it says.
Birne Binegar: Okay, any other questions.
Birne Binegar: Okay, for those who cannot see you should perhaps change your.
Birne Binegar: Your your point of view, there should be at least three windows, let me stop the share here.
Birne Binegar: You should be seeing.
Birne Binegar: My pad of paper world draw things out by hand you'll see my voice so, hopefully, you can read my lips, a little bit, and also the lecture slides that all I do be a screen share but, hopefully with these three modes of.
Birne Binegar: Communication these lectures will be fairly easy to follow.
Birne Binegar: Oh in calculus three also learned vectors using vertical notation well in fact in this course to we will eventually convert to writing vectors vertically, rather than.
Birne Binegar: than horizontally.
Birne Binegar: In fact, when we write them vertically, the idea is actually the subtly think of these vectors as a as a special case of an even more general object called a matrix.
Birne Binegar: On us will introduce those now so a matrix is more than just the list of numbers it's a set of numbers that are written in a particular order but also vertically and horizontally so.
Birne Binegar: Unless this number this list 000 could also be written vertically and that perhaps what you're alluding to my thing vectors in physics are represented vertically.
Birne Binegar: As long as we know, the ordering these things are equivalent as long as this is the first component, the second component third component.
Birne Binegar: The first component second column third component, these are the same doctor as ordered list of numbers it's just i'm using different notation right to do display that ordered list but.
Birne Binegar: When we start to think of vectors as special case of more general objects say 123212 here, I have six numbers but i've got.
Birne Binegar: They occur in a certain order along horizontally, in a certain order overtly So these are what we'll call may trustees and you see that this horizontal, this is a special case of them.
Birne Binegar: matrix with one row and three columns while this vertical vector is a special case of freedom in a matrix with three rows and single call but that's that's all coming up in friday's lecture okay.
Birne Binegar: yeah and yes.
Birne Binegar: You can add vertical vectors just if you would for Donald actors, the as long as you the idea issue and the same component of the same component.
Birne Binegar: So if you're writing kept to disrupt written horizontally, you know you grab from the first lot you combine it with the second floor to get the sun.
Birne Binegar: And that doesn't make any difference whether you write the thing vertically or horizontally, is how you combine the individual component, it gives you the definition of that vector addition.
Birne Binegar: there's a matrix compared with dictionary coding ah somewhat it's actually closer to if you're thinking of coding I would think of it as a two dimensional array.
Birne Binegar: So Ray where you have yeah.
Birne Binegar: You need to have two bits of info to it to indices to indicate a particular element of of the Ray.
Birne Binegar: Okay.
Birne Binegar: Really questions.
Birne Binegar: Okay, good, then I will see you on Friday.