Question

Is the function defined by T(p)=p⋅q a linear transformation, where p and q are polynomials of degree n or less?

a) Yes

b) No

Answer 1

No, the function T(p)=p\(cdot\q) is not a linear transformation since it does not satisfy the homogeneity property if p and q are polynomials of degree n or greater when combined, as the resulting polynomial may exceed degree n.

Step-by-step explanation:

The student asked whether the function defined by T(p)=p\(cdot\q), where p and q are polynomials of degree n or less, is a linear transformation. The answer is no. To be a linear transformation, a function must satisfy two main properties:

1. Additivity: T(p1 + p2) = T(p1) + T(p2) for any polynomials p1 and p2.
2. Homogeneity of degree 1 (Scalar multiplication): T(cp) = cT(p) for any polynomial p and scalar c.

For the given function, if we take two polynomials p1 and p2, we get T(p1 + p2) = (p1 + p2)\(cdot\q)= p1\(cdot\q)+ p2\(cdot\q), which seems to satisfy additivity. However, for homogeneity, T(cp) = cp\(cdot\q) is not equal to cT(p) = cp\(cdot\q), if degree of p and q is n or greater when combined, because the multiplication of two polynomials can result in a polynomial of degree higher than n. Therefore, it does not satisfy the homogeneity property of linear transformations when p and q are polynomials of degree n, as the result can be a polynomial of degree greater than n.