Introduction to eigenvalues and eigenvectors | Linear Algebra | Khan Academy

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Tacitly, we applied an image-conversion (spanning) from Rn to Rn Where it was interesting for us to find vectors that grow mainly through conversions So the vectors that have a shape … the transformation of the direction here is equal to the enlarged image of the vector And if that doesn’t sound familiar, I can go back to memory a little When we were looking for the base vectors for conversion … let me draw it here This was from R2 to R2 R2 to R2 So let me draw R2 here Let me assume I have the vector here …. I will add the vector … and say V1 was equal to vector 1,2 Here we have the lines extending along the vector We had this problem in several previous videos as I had a turn-over conversion So if we call that line L and T is a transformation from R2 to R2 that flips the vectors across this line So the transformation flips and flips the vectors, flips the vectors via I via vectors across L And if I remember that transformation, I would have a random vector that looks like this, say X Hence, the transformation of x looks like this which is reversed through this line This was an X transformation And if you remember that video where we were looking for a change in base, we could at least define the matrix for transformation, at least the alternative rule Then we can define the matrix for transformation in the standard rule The rule we chose was base vectors that the transformation did not change much or that was enlarged by transformation For example, when I took a transformation of V1, when I took a transformation of V1 where it equals V1 Or we can say that V1 conversion is equal to one times V1 So if you just follow this formula, then this is the little formula that you created here The lambs in this case will be 1, of course, the vector in this case is V1 Zoom the V1 conversion by 1 Now if you have the same problem, we have the same vector that we looked at Well, it was the vector it’s the vector … Let’s say it’s the vector v2 And it is … let’s say it equals 2 minus 1 And then if I take that turn, Since he was perpendicular to the line, he was turned like this And that was a somewhat interesting force as well Because converting V2 in this case is equal to what? Only minus v2, would be equal to minus v2 Or you could say that V2 transformation is equal to one negative multiplied by V2 So these were interesting vectors for us because when we knew the new rule for them as a base vector It was easy to determine the transformation of the matrix In fact, it was easy to compute this matrix We will explore more about this in the future, but I hope you realize that these are interesting vectors There were some instances when we found levels to extend along some vectors And then we had another vector that was derailing from the path like this So we were turning things by taking A mirror image across this we are like … well in that transformation These red vectors do not change at all as this element is inverted So your stooges may fit to be good rules Or maybe they fix good base vectors And in fact they are For this reason, and in general, we are interested in the vectors that the transformation has magnified These will not be all vectors, will they? The vector here, this vector X, is just not getting bigger Rather, it has changed and the trend has changed The vectors that are larger may change directly … may transfer From this direction that direction or you may move from here Maybe this is X and then converting X might be an enlarged picture of X Maybe it is The real streak – I think – its streak will not change and this is what we will occupy ourselves with These elements have a special name And they have a specific name as I want to clarify them because they are useful It is not a sports game we play, even though we sometimes fall into that trap But they’re actually useful for setting rules because in those rules it’s easy to find conversion matrices They are more than natural coordinate systems Most of the time, it is easy to calculate the conversion matrices in these rules So it has special names Since any vector that achieves this element present here is called the vector for the transformation T The lambda becomes a multiplication … and this is the intrinsic value associated with the vector So in the example I just gave where the transformation is flipping around this V1 line, the vector is one by two is the vector of transformation that we have That is why one, two vector vectors and their corresponding intrinsic value is one This element is also a vector vector Vector is two, one is negative and is also subjective A wonderful word, but all you mean is the one that was enlarged by conversion It does not change by any means more meaning than scaling factor And its matching self value is one negative If this conversion and which I do not know matrix convert it Forget what it is We actually have defined it since righteousness If this conversion matrix can be represented as a vector product matrix It should be … it’s a linear transformation And then any V check transform … let me say that the conversion of V is equal to the lambda v which will be the same … you know that Since converting V will only be A times V This is also called the auto vectors of A because A is only a representation of a transform So, in this case, this would be the auto vector of A and this would be the eigenvalue associated with the auto vector So if you give me an array that represents a linear transformation, you can also specify this In the next video, we will define a method for identifying these things What I want you to appreciate in this video is that it is easy for you to say: It is easy to say that vectors do not change much. But I want you to know what that meant As it is easy to enlarge or perhaps even flip it Also, their direction or their lines that they originally made do not change The reason she is interesting to us … well One of the reasons why it is interesting for us is that it creates interesting base vectors …. base vectors with conversion matrices that may be more simple mathematically, or work to find better coordinate systems

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