Let Pn denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let r(x) E Pn be a fixed polynomial of degree n. Let f : P2 → Pn+2 be defined by f (p(x)) = p(x)r(x) for all polynomials p(x) E P2. Is fa linear transformation? Let p(x) = a2x² + a1x + ao and q(x) = b2x² + b1x + bo be any two polynomials in P2 and c E R. a. f(p(x) + q(x)) = a2+b2x2+a1+b1x+a0+b0 (Enter a2 as a2, etc.) f(p(x)) + f(q(x))= + b1x+a0+b0rx a2+b2x2+a1 Does f(p(x) + q(æ)) = f(p(x)) + f(q(x)) for all p(x), q(x) E P2? Yes, they are equal b. f(cp(x)) = ca2x2+ca1x+caOrx c(f(p(x))) = a2x2+a1x+aOrx Does f(cp(x)) = c(f(p(x))) for all c E R and all p(x) E P2? Yes, they are equal c. Is f a linear transformation? f is a linear transformation

Let Pn denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let r(x) E Pn be a fixed polynomial of degree n. Let f : P2 → Pn+2 be defined by f (p(x)) = p(x)r(x) for all polynomials p(x) E P2. Is fa linear transformation? Let p(x) = a2x² + a1x + ao and q(x) = b2x² + b1x + bo be any two polynomials in P2 and c E R. a. f(p(x) + q(x)) = a2+b2x2+a1+b1x+a0+b0 (Enter a2 as a2, etc.) f(p(x)) + f(q(x))= + b1x+a0+b0rx a2+b2x2+a1 Does f(p(x) + q(æ)) = f(p(x)) + f(q(x)) for all p(x), q(x) E P2? Yes, they are equal b. f(cp(x)) = ca2x2+ca1x+caOrx c(f(p(x))) = a2x2+a1x+aOrx Does f(cp(x)) = c(f(p(x))) for all c E R and all p(x) E P2? Yes, they are equal c. Is f a linear transformation? f is a linear transformation

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 4EQ

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