1. (10 points) Let T:P3 → P3 be the linear transformation satisfying T(x2 - 1) =...

Question

Question

Answer 1

Answer #1

T is a linear transformation which simply means that it is a function that maps the given inputs to the given outputs.

Since T is a linear transformation we use the following 2 properties of a linear transformation and solve the problem.

1. T(A+B)=T(A) + T(B)
2. T(cA) = cT(A)

Here c is a constant and A,B are polynomials with atmost degree 3(i.e belong to P₃)

Our plan is to find T(x²) , T(x) and T(1) and then using the above properties we convert it into the form of T(ax² + bx + c).

T(2x) = 2T(x) = 4x => T(x) = 2x

T(3x+2) = T(3x) + T(2) = 3T(x) + 2T(1) = 2(x+3) => 3(2x) + 2T(1) = 2x + 6 => 2T(1) = -4x + 6 => T(1) = -2x + 3

T(x² - 1) = T(x²) - T(1) = x² + x - 3 => T(x²) - (-2x + 3) = x² + x - 3 => T(x²) = x² - x

Now, T(ax² + bx + c) = aT(x²) + bT(x) +cT(1)
= a(x² - x) + b(2x) + c(-2x + 3)
= ax² + (2b-2c-a)x + 3c

For any arbitrary real numbers a,b and c T(ax² + bx + c) = ax² + (2b-2c-a)x + 3c

Answer 2

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