Use mathematical induction to prove that if L is a linear transformation from V to W, then L (α1v1 + α2v2 +· · ·+αnvn) = α1L (v1) + α2L (v2)+· · ·+αnL (vn)

Let T: P₂ → R² be the linear transformation defined by T(a₁x² + ₂x + α3) = [3 sas]: T(6x² + 4x + 5) = -2a1 3a2 + 6a3 Ex: 5 8 Ex: 5

Define the linear transformation T by T(x) = Ax. Find ker(T), nullity(T), range(T), and rank(T). 0 -8 2 A 16 0 19 (а) ker(T) (If there are an infinite number of solutions uset as your parameter.) } 16 1 t, -t,t 19 4

-9 V₁ = --[-] and X₂ such that T(x) is Ax for each x. Let x = A = X₁ -2 [3] and let T: R² R² be a linear transformation that maps x into x₁v₁ +X₂V₂. Find a matrix A -8 and V₂ =