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auk411
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Homework Statement
My question doesn't require numerical calculation. It is more about explanation.
Here it is: what does it mean to say there are unique linear transformations?
My textbook says "unique linear transformations can be defined by a few values, if the given domain vectors form a basis." However, that is all it says.
To "explain" here are 2 examples it gives: T(9x+5) = (.1,.2) and T(7x+4) = (.3,.8). This is a unique linear b/c (4a-7b)(9x+5) + (9b-5a)(7x+4) will equal ... [do some foiling] ... = ax +b for all a,b real. (I'm assuming that what the author is saying is that T(a,b) = (4a-7b)(9x+5) + (9b-5a)(7x+4) = ax +b for all a,b real.
[me: here's how lost I am. I'm not even sure why this is called a linear transformation and not linear transformationS. It looks like we are taking one transformation T_1 and doing something to it with T_2. Shouldn't we be asking why T(9x+5) is a unique linear transformation.
Second example: T(2,1) = 4x +5; T(6,3) = 12x +15. Not unique because T(a, b) = 2ax + 5b works, but so does T(a,b) = 4bx + (3a - b).
I assume a and b are arbitrary domain elements. (please correct any assumptions if they are wrong). How does one arrive at the the formulas T(a, b) = 2ax + 5b and T(a,b) = 4bx + (3a - b)? What are these formulas telling me? What about the equation "T(a,b) = (4a-7b)(9x+5) + (9b-5a)(7x+4) = ax +b for all a,b real" tells me that this is a unique linear transformation? Could I just plug in anything to the a and b?
I'm lost. So anything will be helpful.
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