algebra 9

# algebra 9 - Section I Denition of Vector Space 85(the...

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Section I. Definition of Vector Space 85 (the verification is easy). This vector space is worthy of attention because these are the polynomial operations familiar from high school algebra. For instance, 3 · ( 1 - 2x + 3x 2 - 4x 3 ) - 2 · ( 2 - 3x + x 2 - ( 1/2 ) x 3 ) = - 1 + 7x 2 - 11x 3 . Although this space is not a subset of any R n , there is a sense in which we can think of P 3 as “the same” as R 4 . If we identify these two space’s elements in this way a 0 + a 1 x + a 2 x 2 + a 3 x 3 corresponds to 0 B B B @ a 0 a 1 a 2 a 3 1 C C C A then the operations also correspond. Here is an example of corresponding additions. 1 - 2x + 0x 2 + 1x 3 + 2 + 3x + 7x 2 - 4x 3 3 + 1x + 7x 2 - 3x 3 corresponds to 0 B B B @ 1 - 2 0 1 1 C C C A + 0 B B B @ 2 3 7 - 4 1 C C C A = 0 B B B @ 3 1 7 - 3 1 C C C A Things we are thinking of as “the same” add to “the same” sum. Chapter Three makes precise this idea of vector space correspondence. For now we shall just leave it as an intuition. 1.9 Example The set M 2 2 of 2 2 matrices with real number entries is a vector space under the natural entry-by-entry operations.

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