Linear Operator. Need help

[Dimentr]

Let V be the space of quadratic forms in variables t1, t2 over a field K, u ∈ M (2, K), a mapping a: V → V
is given by the equality (a (f)) (t) = f (tu), where t = (t1, t2).
a) Prove that a ∈ End (V).
b) For which matrices u is the operator a diagonalizable?
c) Find Im (a) and Ker (a).

 

[Deleted member 4993]

Let V be the space of quadratic forms in variables t1, t2 over a field K, u ∈ M (2, K), a mapping a: V → V
is given by the equality (a (f)) (t) = f (tu), where t = (t1, t2).

a) Prove that a ∈ End (V).​

b) For which matrices u is the operator a diagonalizable?
c) Find Im (a) and Ker (a).

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