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hawatajwizp 2022-06-20 Answered
Let φ : K [ x ] n K [ x ] n 1 with φ the linear transformation defind by φ ( f ) = f . Select a base and find the matrix of the linear transformation.
I took the standard basis for grade-n polynomials:
B =< 1 , x , x 2 , , x n > , φ ( B ) = φ ( 1 , x , x 2 , . . . , x n ) = ( 0 , 2 x , . . . , n x n 1 )
So, the matrix is [ 0 2 x n x n 1 ] ?
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Answers (2)

Ryan Newman
Answered 2022-06-21 Author has 26 answers
Since φ ( 1 ) = 0, φ ( x ) = 1, φ ( x 2 ) = 2 x, and so on, the matrix is [ 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 n ] .Note that this matrix has n ( = dim K [ x ] n 1 ) lines and n + 1 ( = dim K [ x ] n ) columns.
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Jasmin Pineda
Answered 2022-06-22 Author has 2 answers
I don't really understand why it's constructed like this. From the 2nd column forward every pivot of that submatrix is the coefficient of each polynomial bu, I don't get why the first column is like so.
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