Let (..) be the usual inner product on the complex vector space (₂[x]. Recall that is a 1-dimensional vector Space over C. Consider the function To C₂[x] → C given by T(g(x)) = (g(x), x) A) Prove that T is a linear transformation. B) Find a basis for the kernel of T.

Let V be a vector space, v, u E V, and let T₁ : V → T₁ (v) = 5v +6u, T₂ (v) = 4v 4u, Find the images of v and è under the composite of T₁ and T₂. (T₂T₁)(v) = (T₂T₁)(u) = - V and T₂ : V → V be linear transformations such that T₁(u) = −3v - 3u, T₂(u) = 6v+6u.

Consider the vector space F of all functions from R to R with pointwise addition and scalar multiplication. (a) Show that the three vectors fi = e“, f2 = xe", f3 = x²e" are linearly independent in F. (b) Consider the subspace H = span{f1, f2, f3}. Let T : H → H be the linear transformation defined by df T(f)= + f. dx Find the matrix of T' with respect to the basis B = {fi, f2, f3} of H (i.e., find 8[T]B). (c) Is there a basis C of H in which c |T|c is diagonal? Explain why or why not.

Define T: R² R² by T() = T 21 =[ =[ U2 that T is a linear transformation. ● a) Let u = I1 and 7 = X2 = 3x12x2 2x2 V1 be two vectors in R2 and let c be any scalar. Prove U₂