The standard way, I multiply by A, and I get Ax. Ways of representing the exact same vector, Now D assumes that you have x inĬoordinates with respect to this basis, so with respect Then we would say that D is the transformation matrix for T. To my alternate nonstandard coordinate system. Representation of x times the coordinates of x with respect Represent that dot with this other coordinate system,Ĭoordinates with respect to this basis, it should beĮqual to the product of some other matrix. So that's what that guy is right there- so if I want to Of x in our codomain in coordinates with respect to B. Right here could be some other matrix times thisĬoordinate systems. That's the same thing as mapping from this kind of way What, should always map from that dot to that dot. That thing in nonstandard coordinates if I haveĬoordinates with respect to this other basis here? Well, T should still Matrix A- I then get the mapping of T in standardĬoordinates. Multiplying that thing in standard coordinates times the Transformation T- that's like applying this matrix A to it or That's in standard coordinates, and I apply the So this is an interestingĪn interesting question into your brain. But I could represent it withĬoordinates with respect to my basis just like that. It with coordinates with respect to this basis. Some linear combination of these guys, or you can represent Here, that vector right there, is also in Rn. This would be some other set ofĬoordinates, but it's still representing the same basis. When we represent it in standardĬoordinates, it's just going to be that right Represented with coordinates with respect to this Is another way of saying that any vector in Rn can be Linearly independent and any vector in Rn can be representedĪs a linear combination of these guys, which Just another way of saying that all of these vectors are Let's say that I have some basisī that's made up of n- it has to be linearlyīasis- of n vectors v1, v2, all the way to vn. So that last part I said was aīit of a mouthful, so let me make it a little bit With coordinates with respect to any of those bases. Represent that vector, because Rn has multiple spanningĬan represent Rn, and each of those bases can generate aĬoordinate system where you can represent any vector in Rn I never even said this blue partīefore, because the only coordinate system we wereĭealing with was the standard coordinate system or theĬoordinates with respect to the standard basis. A is the transformation matrixįor T with respect to the standard basis. This A is the transformationįor T only when x is represented in standardĬoordinates, or only when x is written in coordinates with Standard coordinates, or it's being represented with respect Like that, we just assume that it's being represented in It can be represented inĭifferent coordinate systems. Videos, we've learned that the same vector can be represented The word that A is the- we could either call it the matrixįor T, or let's say it's the transformation Transformation, we know that the mapping of x to its codomain To some other member of Rn, which is also the codomain. Our domain, let's call that vector x, then T will map it Is just Rn, then its codomain is also Rn. Transformation T that is a mapping from Rn to Rn. I can throw a flaming ball of monkey poo 1600m or 1 mile or 1.6km or 160,000cm or 5249 feet but youre not converting the same way every time or you'll get the same number each time which is not right. not every unit converts the same, neither does every coordinate system. You don't realize it, but youre changing basis when you convert units because youre changing literally how far apart the tic marks on the ruler (or coordinate axis) are. But if you measured in farenheit, you would need to do something completely different to convert to Kelvin. The thermometer measures in Celsius but you need kelvin to do the calculations properly (just the way PV=nRT works). Think like youre in chemistry, and you need to do something with a gas, like cow farts (just to make this interesting). It's not that you want the result to be the same type of thing, you want the result of the transformation to be invariant, meaning that the object is the same, but the way you represent it might be different.
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