Algebra

Coordinates with respect to a basis | Linear Algebra | Khan Academy

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Let’s say I have a subspace of Rn Let’s call it V as V is a subspace of Rn And let’s say that Group B – they asked that in blue – let’s say that Group B is a base of V with some vectors It has some vectors And let’s say I have v1, v2 through vk If we can see that we have a number of k vectors, then v is Subspace after k That means, if I had a vector a …. let’s say I have A vector is a, which is an element of subspace This means that I can represent a as a linear equation Composed of these symbols here So I can write a as a constant Multiplied by the basic vector plus a constant what is multiplied by The second primary vector And we continue right up to the k from the constants Times the vector in the direction k Now, I have used the term “coordinates” rather loosely Before, but now we have to have a more precise definition I would call these constants here c1, c2, c3 down to Right up to the vector ck I’ll call it all – they asked her in a new color – The coordinates of a relative to the base B For B And we can write it like this And we can write the vector a Our vector a But if we want to write it in coordinates For base set B like this I put it in parentheses and put the base set here What indicates that we will write these coordinates For the base set So I’m going to write this as follows I will put these constants These constants in the linear equation must be found from Base vectors to find a So we have c1 c2 and even ck Well there’s a little interesting Something worth noting here V as the basis for Rn so anything in V is also in Also found in Rn But V has k of vectors It has k dimensions It can be the same size as the value of n, but perhaps It may be less valuable We could have two vectors in R3 and in this case V would be Level in R3 but we can represent that In other dimensions But if you specify something in the subset For its foundation, note that you can have it Many vectors in subspace You just have to have many coordinates So even if a belongs to Rn I can only represent it in the coordinates of k Because basically you give it a place Within, let’s say it was a level, it was within a representative level In the subset Well let’s enable this a little Let’s take a few examples Let’s say we have a subspace Let’s clarify this more Let’s say I have a set of n vectors Suppose v1 is a vector 2.1 And v2 is the vector 1,2 Now you can immediately see that foundation or The set of v1 and v2 foundations for R2 is what that means A vector in R2 can be represented as a linear vector When making a combination of these groups I can do a visual argument Or we can know that R2 is two-dimensional and We can have two simple linear vectors Independents You can see that In fact, it is the simplest way to define this is if you can Take 2,1 and 1,2 and put it in a miniature column combination You will have a symmetric matrix of 2 x 2 You will produce 1,0,01 which you will know These two here are basic vectors So this is a review for that We saw this before But to imagine that Imagine this So if I were to represent it graphically as we would graphically represent it These vectors, how do 1,2 appear? Let me draw pivots here Let’s draw it We’ll do it in another color Let’s say that this is the vertical axis And this horizontal axis 2.1 would look like this And then 1,2 will go up We will go up one cell up So this vector number one here This is 2,1 this is the first vector And then 1,2 would look like it would, rather its shape would be like this I drew it in a standard position 1 and then we go up 2 1,2 It looks like this So when talking about coordinates for To these grounds to consider that we have a R2 I’ll show it so that you can easily find a linear vector Linear vehicle Well suppose we have 3v1 + 2v2 What might that equation be equal to? Will equal its vector Well, 3 x 2 is 6 plus 2 x 1 So it’s vector 8 and then i have vector 3 times v1 Plus 2 times that So 8.7, right? 3 + 2 x 2 is for 8.7 So if we were to just draw 8.7 in the traditional way We will go to 1.2.3.4.5.67.8 and then We go up to 1.2.3.4.5.6.7 Now we have another direction. I will not draw it Here, but I’ll set this point here That could be the point If these are shown as coordinates So we’ll see this point as 8.7 right here Maybe I will write it like this This point is 8.7 If I wanted to draw a ray in the normal position I’ll draw a vector that ends here exactly Now we have this basis here, the base here. Basis B Represented by these vectors here This is B1 and B2 And what we want to do is to represent this Let’s say this is a vector Let’s call it the vector a as a is 8.7 Now we know that if we want to represent the vector a over it Linear component of this basic vector will be 3v1 + 2v2 3v1 + 2v2 So with the knowledge of what we saw here In an earlier part of this recording Vector a for B – they color it the same way as B- For base B, these constants will be equal On these base vectors, where 3 and 2 are equal And let’s see if we can represent this by looking to understand if it is Logically We considered it to be in a new coordinate system Since this vector can be represented as 3,2, this way we can think With a new coordinate system In this system we segmented the vehicle so that it is in the horizontal And therefore it was the first coordinate And then we divided it so that it is in the vertical axis And that was the second coordinate Now in our system, what is the first coordinate? The first coordinate will be a multiple of v1 This v1 or this v1 if it’s going to be a multiple of v1 If this is 1 x v1 Then if we multiply v1 by 2, it will result What is the output of 2 x v1? The output will be 4,2 3 times v1 gives 6,3 Now let’s see 1,2,3,4,5,6,7,8 So it is 6 and then 3 as it is here And the result of 4 times v1 will give us 8 and 4 So you can imagine what I draw here It is the axes, they are the primary axes that are set up by v1 So I can draw it, by looking at it in blue, So you can imagine it like this This would be a straight line like this Then the coordinates tell us how much v1 we have So I’m going to dismantle the coordinate system like this Instead of representing 1 elements I will Representing elements from v1 I will write it like this And while it continues, 9,10, we will represent 5 more points something like that Now, the second coordinate tells us with elements from v2 So this is our first component of v2 and then Our second component, going up to 4, would be 4, 2 So This will be 6 and 3 It will be exactly that It will weigh 6 and 3 It will look something like this If you want to consider it as OK You should consider it a coordinate system Then we have this skewed interstitial paper so that any point You can represent it in v1 direction In some quantity and then you go in the direction of v2 as much as Let me draw that in this graph paper So I can draw another version of v2 exactly like this Just all the multiples of v2 I can move it here like this I can do another tool like this I can do another This is not regular enough I can do that I can work … well I guess you got the idea Let’s make this more tidy This could be more practical with another tool Then I can represent all multiples of v1 like this I’m making a graphic sheet here So that looks something like this It will look something like this And you can imagine the diagonal diagram If you spray it all over Green and this is blue So in our coordinate system, we see 3,2 Which means 3 times the first direction Which happens to be v1 Not the horizontal vehicle anymore It is the direction of v1 So we count 1, 2, even 3 so And then 2 in the direction of v2 So we’re going to count 1, 2, in the v2 direction The result will be here You can imagine things going like this 3 in v1 and 2 in v2 And then we get the point Or you can go on v2 v1 after, but in no way will you return to the original point So that the location vector With vectors 8.7, it is easy to follow and easily On the new coordinate system with coordinates 3.2 Because we calculated 3 times v1, and then we combined it with 3 times v1 It will take us in a direction We’ll walk 3 knots in the direction of v1 and then Two nodes in the direction of v2 This is why it was called coordinates We literally ask how many spaces in v1 V1 direction to go and then how many spaces In the direction of v2 to go But this could, I think, lead us to a clear question We did not use the coordinates before As I might say in previous times I used to say it the whole time Let’s say I have a vector b that is equal I don’t know let’s say it equals – I will represent it as R2 but it is easy to explain Let’s say it Equals 3, -1 (3 and minus 1) If we were to draw that would look like Something like that We’ll go 1,2,3 and then 1 down So something will look like this The point will clarify But why do we call 3 and minus 1 coordinates Why haven’t we done that before? Before we learn linear algebra We called these coordinates since we first started Learn to represent graphically Why do we call these coordinates Or how can these coordinates be connected? These coordinates are for a base Hassan. These are the coordinates for Basas These are actual coordinates of The standard basis If you can imagine, let’s see the standard basis In R2 it looks like this We can have E1 which is 1.0 and we have e2 which is 0.1 This is just just a standard basis conversion in R2 Then we can say that s is equal to the set of e1 and e2 Then we can say that s is the standard basis for R2 And it is because these two vectors here are orthogonal This is 1 in the horizontal vehicle And this is 1 in the vertical spacecraft And any vector in R2 – say that I have a vector x, y in R2 – is equal to x times e1 plus y times e2 Then we can say that if you want to write a vector x, y, if you want to write for this normative basis here, then the coordinates are equal to the definition that we did earlier in this video about the basic vectors here. And these weights are on the e1 and e2 compounds, so it will equal, well the weight here is x and the weight here is y So these coordinates that we talked about from the beginning, these are coordinates that really are consistent with our definition of vectors in this video But maybe we can be a little bit more accurate We can now call it coordinates relative to the standard basis Or we can call these here in standard coordinates I wanted to point this out This seems somewhat easy or somehow obvious But I wanted to clarify our old use of the word “coordinate” that was not compatible with this new definition of coordinates as weights on some basic vectors. Because even when we were using the previous coordinates, or the old way we used them, these were actually loads on the standard basic vectors

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