Question

Generalize help
P is polynomials, prove T is a linear transformation
T(p(x))=p(x^2)
I can do in particular one, but not know how to make it in general
please, guide me

Answer 1

p(x1) =x+2 --> T(p(x1)= x^2 +2
p(x2) = x^3+2x -->T(p(x2) = x^6+2x^2
T(p(x1)+p(x2))= T(px1)+T(px2)
adding a proof of scalar multiplication, I can get T is a linear transformation
But I need a general form and don't know how to do

Answer 2

@wio

Answer 3

Well, \(p(x)\) is allegedly a polynomial. \[
p(x) = \sum_{k}a_xx^k
\]

Answer 4

yes

Answer 5

a_k x^k, typo, right?

Answer 6

How?

Answer 7

Here is one:\[
T(cp(x)) = cp(x^2) = c[p(x^2)] = cT(p(x))
\]

Answer 8

I mean p(x) = sum a_k x^k

Answer 9

\[
T(p_1(x) + p_2(x)) = p_1(x^2)+p_2(x^2) = T(p_1(x)) + T(p_2(x))
\]

Answer 10

how can we go from left to middle one?

Answer 11

Let \(p_3(x) = p_1(x)+p_2(x)\)
Then \(p_3(x^2) = p_1(x^2)+p_2(x^2)\)

Answer 12

oh yeaaaah. Thanks a ton