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1. Let V be the vector space consisting of all polynomials of degree < 3 with real number coefficients. Let f : V →V be the function defined by f(p(x)) = x²p(0) + xp(1) + p(2). %3D (a) Prove that f is a linear map. (b) Find a matrix representation for f using the basis of polynomials {1,x, x², x³} for the input space and for the output space.

1. Let V be the vector space consisting of all polynomials of degree < 3 with real number coefficients. Let f : V →V be the function defined by f(p(x)) = x²p(0) + xp(1) + p(2). %3D (a) Prove that f is a linear map. (b) Find a matrix representation for f using the basis of polynomials {1,x, x², x³} for the input space and for the output space.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Let V be the vector space consisting of all polynomials of degree < 3 with real number coefficients. Let f :V → V be the function defined by f(p(x)) = x²p(0) + xp(1) + p(2). %3D (a) Prove that f is a linear map. (b) Find a matrix representation for f using the basis of polynomials {1, x, x², x³} for the input space and for the output space. 2. Let f : R³ → R³ be the function in which f(v) is the result of first projecting v orthogonally onto the plane 3r – 4y+12z = 0 and then rotating the resulting vector 45° in that plane counter-clockwise with respect to the direction vector 3 d = |-4 12 Find a matrix B such that f(v) = Bv. %3D
Transcribed Image Text:1. Let V be the vector space consisting of all polynomials of degree < 3 with real number coefficients. Let f :V → V be the function defined by f(p(x)) = x²p(0) + xp(1) + p(2). %3D (a) Prove that f is a linear map. (b) Find a matrix representation for f using the basis of polynomials {1, x, x², x³} for the input space and for the output space. 2. Let f : R³ → R³ be the function in which f(v) is the result of first projecting v orthogonally onto the plane 3r – 4y+12z = 0 and then rotating the resulting vector 45° in that plane counter-clockwise with respect to the direction vector 3 d = |-4 12 Find a matrix B such that f(v) = Bv. %3D
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