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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
• These linear transformations are probably different from what your teacher is referring to; while the transformations presented in this video are functions that associate vectors with vectors, your teacher's transformations likely refer to actual manipulations of functions.

Unfortunately, Khan doesn't seem to have any videos for transformations, reflections, etc. in his algebra playlist, but the links below might be useful.

WolframMathWorld: http://mathworld.wolfram.com/Transformation.html

Paul's Online Notes: http://tutorial.math.lamar.edu/Classes/Alg/Transformations.aspx

Hope this helps and good luck!

• Is another name for this 'linear mappings'?
• Linear Transformation = Linear Mapping = Linear Function
• 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?
• The textbook definition of linear is: "progressing from one stage to another in a single series of steps; sequential." Which makes sense because if we are transforming these matrices linearly they would follow a sequence based on how they are scaled up or down.
• 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?
• But how would we get a scalar like 1.1 from just adding a vector with itself, or pi for that matter?
This is a great question, and one I used to ask myself. Ultimately, there examples of transformations that satisfy vector addition, but not scalar multiplication, so both conditions for linearity are in fact necessary.
• 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?
• Let v be an arbitrary vector in the domain. Then T( 0 ) = T( 0 * v ) = 0 * T( v ) = 0. So you don't need to make that a part of the definition of linear transformations since it is already a condition of the two conditions.
• 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?
• No. Knowing the kernel tells us which basis vectors are sent to 0, but the remaining basis vectors could still be sent anywhere.