Algebra 819 - Homework 2 Samuel

1 Let K be a field, V a finite-dimensional vector space over K, and T ∈ EndK(V ). For k ∈ K and S ∈ EndK(V ), define kS : V → V, v → k S(v). (a) Let Φ : K[x] → EndK(V ) be defined by Φ(k0 + k1x + k2x + . . . + knx) = k0T 0 + k1T 1 + k2T 2 + . . . + knT, where T 0 = idV . This is well-defined because a polynomial is completely determined by its coefficients. We will show that this is a homomorphism with Φ(k) = k idV and Φ(x) = T . First, observe Φ(k) = Φ(k + 0Kx + . . . + 0Kx) = k T 0 + 0KT 1 + . . . + 0KT = k idV , and Φ(x) = Φ(0K + 1Kx + 0Kx + . . . + 0Kx) = 0KT 0 + 1KT 1 + 0KT 2 + . . . + 0KT = T. So we have verified the two required images under Φ. Second, let us check that Φ is a homomorphism between additive groups. Let f, g ∈ K[x] be of degree m and n, respectively. Without loss, assume n > m. Then