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For each non-negative integer m, and each field K, let K[2] denote the vector space over K consisting of the polynomials in z with coefficients in K. Let K[x]m denote the subspace of K[x] consisting of the polynomials in x with coefficients in K and of degree

For each non-negative integer m, and each field K, let K[2] denote the vector space over K consisting of the polynomials in z with coefficients in K. Let K[x]m denote the subspace of K[x] consisting of the polynomials in x with coefficients in K and of degree

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CR: Review Exercises
Problem 47CR: Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
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For each non-negative integer m, and each field K, let K[2] denote the vector space over K consisting of the polynomials in z with coefficients in K. Let K[x]m denote the subspace of K[x] consisting of the polynomials in x with coefficients in K and of degree <m. (d) Let D be the map from R[x]s to itself that takes each polynomial f(x) to its derivative f'(x). (i) Prove that D is a linear map on R[*]3. (ii) Write down the matrix representing D with respect to the basis (1, 2, 2², 2³) of R[2]3. (iii) Compute the rank of D and the nullity of D.
Transcribed Image Text:For each non-negative integer m, and each field K, let K[2] denote the vector space over K consisting of the polynomials in z with coefficients in K. Let K[x]m denote the subspace of K[x] consisting of the polynomials in x with coefficients in K and of degree <m. (d) Let D be the map from R[x]s to itself that takes each polynomial f(x) to its derivative f'(x). (i) Prove that D is a linear map on R[*]3. (ii) Write down the matrix representing D with respect to the basis (1, 2, 2², 2³) of R[2]3. (iii) Compute the rank of D and the nullity of D.
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