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.
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
Parts (2), (3), (5), (6)
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