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Discussion in 'Mathematics' started by jnboy1, Jun 29, 2011.
Gosh, have I stumped everyone or is it so obviousand noone dare tell me.......
The first thing you need to do is look up the correct definition of a linear transformation on a vector space. Your definition is incomplete and slightly odd in that your transformation appears to be acting on a vector space of matrices - not the first thing that comes to mind. Get that right and then someone might be able to answer your question.
The definition I have given is from the FP1 book, I am not sure this editor will put it in the form it was originally written but I will give it a go.
A linear transformation T has these properties.
1. T(kx ky) = kT(x y), where k is a constant
2. T[9x1 y1) + (x2 y2)] = T(x1 y1) + T(x2 y2)
Obviously (x1 y1) are meant to be vectors and written vertically.
This is all I had to work with at the start.
I have tried to find info from other sources but not really had much success.
Perhaps you would be kind enough to provide the complete definition from where I could start.
Do you mean if T(X) = A.X where both A & X are matrices? Dot standing for matrix multiplication?
You would get
AX + AX = A[X+X] statement 1
and kAX = AkX statement 2
If transformation T is not linear then it either cannot be represented as a matrix A or one of statements 1 or 2 you would be able to find a counter example?
Let me try that last bit again...
If transformation T is not linear then
either T cannot be represented
as a matrix A
one of statements 1 or 2 would be false i.e. you would be able to find a
Is that what you are trying to get at?
AX means matrix muliplication for the foamy, frothing at mouth, ranters on here.
Thanks for you interesr afterdark
T is a transformation such that
x goes to 2x + 1
Y goes to y + 4
and (x) is apoint say (1,0) agaun written vertically as a 2 by 1 matrix
(god for a decent editor)
i get the two conditions in terms of how they work just not sure why a counter example shows that the transformation is not linear.
Because the two properties define a linear transformation
If x is a point, as you say, what is 2x+1?
Amongst other things, a linear transformation has to send zero to zero. Your transformation doesn't seem to do that.
Isn't that the counter example.
For it to be a linear transformation
1) the transformation only involves linear expressions of x and y
2) the origins not moved by the transformation
3) and 4) as per your post number 4 ( ignoring the 9 which is a typo)
If your transformation does not satisfy each of these conditions then it is not a linear transformation
Again thanks for the input. Yiu have clarified the question for me. I suppose what i am really asking is why are those conditions the definition of a linear transformation?
It comes from the definition of linear in linear algebra where the operations of addition and multiplication are preserved.
Compare with affine transformation.
Your given T is affine.
And that is why i love this forum!
Thank you very much for your time and input everyone.
I now understand clearly and will be able to pass that on to my class.
The conditions for a map from one vector space to another to be linear can be encapsulated in one requirement:
T: V -> W is linear if T(ax + by) = aT(x) + bT(y), for all x, y in V and a, b in the underlying field of scalars (probably the reals in the case under consideration).
Sometimes this single condition is deconstructed and written as several conditions - I think this is where the OP was coming from.
Thinking in terms of linear transformations geometrically, they have to preserve the origin, but can be composites of rotations, reflections, scale transformations.
The OP's transformations look more like affine transformations, as noted by afterdark, which are effectively a linear transformation followed by a translation.