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# Linear transformations

Introduction to linear transformations. Created by Sal Khan.

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• When my teacher says to learn Transformaions: reflections, rotations, translations, dilations.... is this the video I should be watching for that or is Linear transformations something different? if it is could you tell me what that video is called so I can look it up? Thank you so much.. confused a bit here =P • Is another name for this 'linear mappings'? • Simple question, (apologies if answered, I'm about 1/2 way through), but, what exactly does "Linear" mean. 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?

Also, can you name a condition or two where 'linearity', that is, the criterion will consistently broken?

I hope I'm clear on the type of answer I'm looking for. Thanks,
TB

EDIT: 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 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.

I guess I answered my own question =D

You mentioned squares and exponents. Curious, something inherent in either transforming or adding either squares or exponents is causing a loss of information.

Care to take this logic further? • Why do we need to have two conditions here?

Isn't the vector addition enough? After all, if you can add vector a and a scalar times vector a, then this is the same thing as just multiplying the vector by that scalar + 1, isn't it? • It would be good if there were more practice problems and quizzes on this unit. It is hard to keep track of all this information without applying it. • In order for it to be a linear transformation doesn't zero vector have to satisfy the parameters as well? If it is how come it wasn't in the video? • At , Sal said that component of a vector is scalar..but component of a vector also have their direction (like component along x axis or so)..right? so in that way component of a vector should also be vector, i think...! Well, m confused..plz help..and sorry for the silly out of context question.... :) • Well, strictly speaking component of a vector, that is just what is written inside a vector cell is a scalar, it has no information in which cell it was written. What you are talking about is vector decomposition, i.e. representing a vector as a sum of axis-aligned vectors, consider example, given vector

v = (3, 4, 5)

it has scalar components - just numbers 3, 4, and 5
it could also be decomposed into a sum, like this

vx = (3, 0, 0)
vy = (0, 4, 0)
vz = (0, 0, 5)
v = vx + vy + vz

look up 'vector basis'
• Is there a third property of a transformation being linear: T(0) = 0? I can't think of when this wouldn't be the case, unless there's a constant in the transformation without a variable.. Wanted to confirm if this is a property or not... Thanks. • Sal can we find a linear transformation by knowing the basis of its kernal? • At , Sal mentions that if you're dealing with a linear transformation that involves only a linear combination of different components of inputs, you're "probably" dealing with a linear transformation. But if we're talking about a "linear combination" of components, wouldn't it ALWAYS be a linear transformation?? If not, can someone give an example where a linear combination of components leads to a Non-linear transformation?
(1 vote) • 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. 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. Similarly, any combination of constants results in one bigger constant. This means that we can , by proving that T(vector a) = [c1*a1, c2*a2, ..., cn*an] is L.T. for any vector a and for any series of constants c1, c2, ... cn, prove that any transformation including only linear combinations is a transformation that is L.T.
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).

I made a small mistake by first not seeing that linear combination ONLY involves vectors. Constants are out of the question, therefore I should not have spoken about "linear combinations of vectors >>and constants<<".

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.