Question
Let T : P2 → P1 be defined by
T (a + bx + cx2) = b + 2c + (a − b)x.
Check that T is a linear transformation. Find the matrix of the transformation with respect to the ordered bases B1 = {x2, x2 + x, x2 + x + 1} and B2 = {1, x}. Find the kernel of T. .
Answer :
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Skip in 10
Let's go step by step to solve this problem: ### Part 1: Check if \( T \) is a linear transformation We have the map: \[ T(a + bx + cx^2) = b + 2c + (a - b)x \] We need to check if \( T \) satisfies the two properties of linearity: 1. Additivity: \( T(u + v) = T(u) + T(v) \) for all \( u, v \in P_2 \). 2. Homogeneity: \( T(\lambda u) = \lambda T(u) \) for all scalars \( \lambda \) and \( u \in P_2 \). Let's first check additivity: Let \( u = a_1 + b_1x + c_1x^2 \) and \( v = a_2 + b_2x + c_2x^2 \). Then, \[ T(u + v) = T((a_1 + a_2) + (b_1 + b_2)x + (c_1 + c_2)x^2) \] \[ = (b_1 + b_2) + 2(c_1 + c_2) + [(a_1 + a_2) - (b_1 + b_2)]x \] \[ = (b_1 + 2c_1 + (a_1 - b_1)x) + (b_2 + 2c_2 + (a_2 - b_2)x) \] \[ =
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