Start with how the definitions of P2 and P3.derryck1234 said:Homework Statement
Let T: P2-P3 be the linear transformation defined by T(p(x)) = xp(x). Which of the following are in R(T)?
(a) x2
(b) 0
(c) 1 + x
Homework Equations
R(T) is the the set of all vectors in P3 which are images under T of vectors in P2.
The Attempt at a Solution
I just don't know where to start? I don't know what p(x) is?
In mathematics, the range of a function refers to the set of all possible output values when the function is applied to different input values. In the context of T (polynomial), the range refers to the set of all possible values that the polynomial can produce.
The range of T (polynomial) can be calculated by finding the roots of the polynomial and identifying the highest and lowest values that the polynomial can produce. This can be done by graphing the polynomial or by using calculus methods such as finding the derivative and determining the critical points.
Yes, the range of T (polynomial) can be infinite if the polynomial has no real roots and has an increasing or decreasing trend. This means that there is no upper or lower bound to the values that the polynomial can produce.
The degree of the polynomial, which is the highest power of the variable, can greatly affect the range of T (polynomial). Higher degree polynomials have more complex shapes and can produce a wider range of values compared to lower degree polynomials.
Yes, the range of T (polynomial) can include negative values if the polynomial has negative coefficients or if it has real roots that are negative. It is important to consider both positive and negative values when determining the range of T (polynomial).