The discussion revolves around identifying which of the provided functions are linear transformations in the context of linear algebra. The functions in question are L(x,y,z) = (0,0), L(x,y,z) = (1,2,-1), and L(x,y,z) = (x^2 + y, y - z).
Some participants have begun to explore specific examples to test the linearity of the transformations, particularly focusing on the first function. Others express confusion about the application of the transformation definitions and seek clarification on how to approach the problem.
Participants note a lack of understanding regarding the transformations themselves and how they relate to the linearity conditions. There is mention of using specific vectors to test the properties, but no consensus has been reached on the outcomes for all functions.
Start with a couple of vectors, say u = <x1, y1, z1> and v = <x2, y2, z2>.superdave said:Homework Statement
which of the following are linear transformations.
a) L(x,y,z) = (0,0)
b) L(x,y,z) = (1 ,2, -1)
c) L(x,y,z) = (x^2 + y, y - z)
The Attempt at a Solution
I know that L is a linear transformation if L(u + v) = L(u) + L(v) and L(ku) = kL(u).
I am not sure how to apply that to those equations.
Please help.
Mark44 said:Start with a couple of vectors, say u = <x1, y1, z1> and v = <x2, y2, z2>.
For a), see if L(u + v) = L(u) + L(v) and L(ku) = kL(u). Do the same for b) and c).
Office_Shredder said:Well if it was L(v)=Av it wouldn't be interesting to ask if the transformation was linear!
For example for the first one if (x,y,z)=(1,2,4)
L(1,2,4)=(0,0).
And if (x,y,z)=(1,1,1)
L(1,1,1)=(0,0)
So L(1,2,4)+L(1,1,1)=(0,0)+(0,0)=(0,0).
Is this equal to L(2,3,5)? ((1,2,4)+(1,1,1))