Question

let T:P2 to P3 be linear transformation given by formula T(p(x))=xp(x) then which of the following vectors are in Range(T)
a) x+x^2
b)1+x
c)3-x^2

Answer 1

well, if i have a polynomial in P2, say p(x) = a+bx+cx^2, then when i do the transformation I get:
T(p(x)) = x(p(x))= x(a+bx+cx^2)=ax+bx^2+cx^3 (there is no constant term)
As you can see, there is no possible way to have a constant if you run a polynomial through that transformation. So the answer must be a., its the only answer choice that oesnt have a constant in it.

Answer 2

doesnt*

Answer 3

sir,why the choice b and c are not in range(T) can you please explain sir

Answer 4

sir explain

Answer 5

choice b and c arent in the Range(T) because they arent in the form ax+bx^2+cx^3. answer choice b has a 1 in it, answer choice c has a 3. Those are constants. There is no way a polynomial could have gone through the transformation and kept its constants. Its sorta the same reasoning for why -2 isnt in the range of x^2 (talking about real numbers only). Its because there is no real number squared that could be -2.