B obtain a basis for w hint use the coordinate

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(b) Obtain a basis for W. (Hint: Use the coordinate mapping with respect to the natural basis for P 2 to obtain an equivalent problem in 3 . Solve that problem and translate the results to the polynomial space. Observe that the linear equation you have to solve is the one giving the defining condition on the members of W.) 7. (7 pts.) Let T: P 2 P 2 be the function defined by the rule T(p) = p - 2p , where the primes denote differentiation with respect to t. (a) Verify that T is a linear transformation. (You need only quote the appropriate properties of differentiation.) (b) Obtain the matrix [T] B , where B = {1, 1 + t, 1 + t + t 2 } is an off the wall basis for P 2 . (c) What can you tell about the dimensions of the kernel and the range by using the matrix [T] B ??
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