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.
So can someone explain what a unique linear transformation is?
To help make this question more manageable, let me give you some examples from the text:
1) $T(9x+5) = (.1,.2)$ 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.
Q: mmm, so what about $(4a-7b)(9x+5) + (9b-5a)(7x+4)$ will equal ... [do some foiling] ... = $ax +b$ for all $a,b$ real' tells me this is a unique linear transformation? Even more fundamentally, just looking at the example problem it seems we have 2 transformations. So if we were going to look for a unique transformation shouldn't we only look at $T(9x+5)$ and/or $T(7x+4)$ individually. Or if we wanted to look at them both should we be asking if a new (unlisted) transformation $T^*$ (which is $T(9x+5+7x +4)$) is a unique linear transformation? Perhaps, I'm just not really clear on what exactly we are finding the linear transformation of.
2) $T(2,1) = 4x +5$; $T(6,3) = 12x +15$. Not unique because $T(a, b) = 2ax + 5b$ works, but so does $T(a,b) = 4bx + (3a - b)$.
Q: Again, what specifically are we finding the linear transformation of? Where does $2ax + 5b$ and $4bx + (3a - b)$ comes from? How did the author arrive at it? I'm just not sure on how to interpret these equations?
That's it.
Yes, I know my questions were probably confused (for I am). But I'm looking for anything to help give me clarity. For I have nothing more to go on that the 2 sentences from the book: "Unique linear transformations can be defined by a few values, if the given domain vectors form a basis. Otherwise it could fail the uniqueness condition, or worse yet, fail the definition of a function."
Thanks in advance.