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Rn degrees rm translation geometry12/29/2023 Rules to figure out if some actual transformations Transformation then this should be equal to c times so let me just multiply vector a times some scalar The transformation of any scaled up version of a vector That's my first conditionįor this to be a linear transformation. If I add them up first, that'sĮquivalent to taking the transformation of each of the Transformation if and only if I take the transformation of Transformation if and only if the following thing is true. The following two things have to be true. I'm being a little bit particular about that, although Rn to rm - It might be obvious in the next video why And a linear transformation,īy definition, is a transformation- which we We already had linearĬombinations so we might as well have a linear Something called a linear transformation because we're Transformation called a linear transformation. Transformation is, so let's introduce a special kind of I hope that shows that the proof is quite simple, so it is certainly not impossible, even quite easy, to state that all examples will work, as there is a simple proof covering every possible linear combination without loss of generality, by making a few simple but key (and easily forgotten) lemmas and assumptions. Constants are out of the question, therefore I should not have spoken about "linear combinations of vectors >and constants<<". I made a small mistake by first not seeing that linear combination ONLY involves vectors. To make the "if T consists of linear combinations of vectors and constants" an "iff ~", all we need to do now is to prove that for any non-linear combination of vectors, but not constants (c1*c2 is still C and thus does not give a different result about whether T is L.T., whatever that result may be). cn, prove that any transformation including only linear combinations is a transformation that is L.T. for any vector a and for any series of constants c1, c2. This means that we can, by proving that T(vector a) = is L.T. Similarly, any combination of constants results in one bigger constant. Simply put (just to explain the concepts of what would need to be included in the proof), we know that any combination of vectors can be expressed as another vector. Unfortunately LaTeX does not work in these comment boxes, as otherwise I could have shown you my proof that any transformation consisting of linear combinations is also a linear transformation. Curious, something inherent in either transforming or adding either squares or exponents is causing a loss of information. I guess that something would be lost in transformation, not addition, so if information is lost in transformation then it would still be lost when they are then added together thus giving a different. Thanks,ĮDIT: With a little inductive reasoning, it appears that if a translation is NOT linear, something is being lost or gained either when either the vectors are added together and then transformed, or something is lost or gained when they are transformed then added together. I hope I'm clear on the type of answer I'm looking for. I understand that it meets those three criterion, but say, in a very abstract sense (and hopefully in laymen's terms), what does it mean? Perhaps it implies continuity? Perhaps it means the transformation won't enter the domain of complex numbers?Īlso, can you name a condition or two where 'linearity', that is, the criterion will consistently broken? DS (font: TiRo 10.Simple question, (apologies if answered, I'm about 1/2 way through), but, what exactly does "Linear" mean.
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