6. We are going to find a basis of all the polynomials in P4 that have 3 ,5 and 7 as a root. How? (1) Define a function T : P4 → R³ by setting T(p(x)) = |P(5) (2) Prove T is linear. (3) Find MT, the matrix of the transformation T. (4) Find a basis of Null(MT). (5) Use your previous answer to find a basis of Ker(t). (6) Explain why the basis of Ker(T) answers the question.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Parts (2), (3), (5), (6)

6. We are going to find a basis of all the polynomials in P4 that have 3 ,5 and 7 as a root. How?
(1) Define a function T : P4 → R³ by setting
[p(3)
T(p(x)) = |p(5)
[P(7)
(2) Prove T is linear.
(3) Find MT, the matrix of the transformation T.
(4) Find a basis of Null(MT).
(5) Use your previous answer to find a basis of Ker(t).
(6) Explain why the basis of Ker(T) answers the question.
Transcribed Image Text:6. We are going to find a basis of all the polynomials in P4 that have 3 ,5 and 7 as a root. How? (1) Define a function T : P4 → R³ by setting [p(3) T(p(x)) = |p(5) [P(7) (2) Prove T is linear. (3) Find MT, the matrix of the transformation T. (4) Find a basis of Null(MT). (5) Use your previous answer to find a basis of Ker(t). (6) Explain why the basis of Ker(T) answers the question.
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