Some vector x, where the vector looks like Might encounter - you could write it a transformation of It's good to see all the different notations that you Let's say it maps to, so this will be equal to, or it'sĪssociated with x1 plus x2. So if you give it aĢ-tuple, right? Its domain is 2-tuple. 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 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? Simple question, (apologies if answered, I'm about 1/2 way through), but, what exactly does "Linear" mean.
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