Final answer:
Vector v is represented in base T coordinates, and to determine v in the standard base, we need to express it in terms of T's basis vectors and then simplify.
Step-by-step explanation:
The student's question pertains to linear algebra, specifically to the change of coordinates between two bases. The vector v is given in the base T, and the goal is to express it in the standard base. To find the vector v in the standard base, we first need to express the bases S and T in terms of the standard base vectors, and then use the coordinates of v in base T to find its standard coordinates.
The bases S and T for P₁ are given by S = (t+3, t-4) and T = (t+1, t-3), respectively. The vector v in base T is [v]ₜ = [-2, 4]. In terms of base T's vectors, v = -2*(t+1) + 4*(t-3). Simplifying this expression will give us the actual vector in terms of t, which can then be compared to the standard base vectors.