Question

determine whether the followin function is linear transformation .
T:f(negative infinty,infinity)->f(negative infinity,infinity) where T(f(x))=1+f(x)

Answer 1

I think it must be this T(f(x)),1+f(x)).
Isn't it?

Answer 2

T(f(x))=1+f(x)

Answer 3

two criteria to check...

Answer 4

\[T(c\vec u)=cT(\vec u)\]and\[T(\vec u+\vec v)=T(\vec u)+T(\vec v)\]

Answer 5

if both these things are true it's a linear transformation, on at least one is not, it aint

Answer 6

what does the transformation to to the function?

Answer 7

...in this case I mean

Answer 8

according to my understanding,is a structure preserving map called hormomophism

Answer 9

yeah, but those names won't help you if you don't understand how to check them

Answer 10

what does T do to the function f(x) ???

Answer 11

transform a function

Answer 12

how exactly does *this* transformation alter the function?

Answer 13

it alters a function by the above two conditions u mentioned,ie if it meets the two conditions it a a linear transformation which transforms a function

Answer 14

you are not understanding, and hence over-complicating my question

Answer 15

you are told that the transformation you have is\[T(f(x))=f(x)+1\]so this transformation takes a vector (in this case a function) and adds 1 to it

Answer 16

so given that the operation of this transformation is to add 1 to the input, what is\[T(cf(x))\]where c is some constant

Answer 17

yah,continue maybe i will understand

Answer 18

look, what is the input of the transform I just wrote above?

Answer 19

negative infinity and infinity

Answer 20

no, it's the vector; the part in the parentheses in the T...

Answer 21

for\[T(f(x))=f(x)+1\]the input is f(x) and the transformation T is to add 1 to it

Answer 22

so what is the input in\[T(cf(x))\]

Answer 23

cf(x)

Answer 24

right, now apply the transformation and what do you get?

Answer 25

cf(x)+1

Answer 26

exactly, so now we know that\[T(cf(x))=cf(x)+1\]now we need to see if that is the same as\[cT(f(x))\]can you check that?

Answer 27

is nt the same since it gives c(f(x)+1)=cf(x)+c

Answer 28

exactly :)
the only c that would make this linear is c=1, which we cannot assume, so the transformation is not linear
you could also have seen that if you tried the other test as well

Answer 29

lets try the 2nd condition

Answer 30

ok, what is\[T(f(x)+g(x))\]?

Answer 31

(f(x)+g(x))+1

Answer 32

exactly, and what is\[T(f(x))+T(g(x))\]?

Answer 33

(f(x)+1)+(g(x)+1)

Answer 34

which simplifies to...?

Answer 35

f(x)+g(x)+2

Answer 36

so is\[T(f(x)+g(x))=T(f(x))+T(g(x))\]?

Answer 37

no

Answer 38

so it does not fit either condition for a linear transformation, though as I said we were done once we say it did not meet at least one criteria

Answer 39

thanks a bunch

Answer 40

welcome!