4. Recall that P2 is the vector space of polynomials with real coefficients and with degree less than or equal to 2 (equipped with the natural operations on polynomials). Consider the linear map L: P2 → P2 defined by its action on the canonical basis В -< х?, х, 1 > as follows: L(x²) = -4x² + 6x, L(x) = -4x + 3 and L(1) = -4. Lastly, let p be the polynomial given by p = 2x +1 – x². (a) Find the matrix representation H = Repg,B(L) of the map h with respect to the basis B. (b) Find the column vector representation, Repâ(p) or [p]g, of the polynomial p with respect to the basis B. (c) Use the matrix obtained in part (a) and the result of part (b) to find the column vector [L(p)]g.

4. Recall that P2 is the vector space of polynomials with real coefficients and with degree less than or equal to 2 (equipped with the natural operations on polynomials). Consider the linear map L: P2 → P2 defined by its action on the canonical basis В -< х?, х, 1 > as follows: L(x²) = -4x² + 6x, L(x) = -4x + 3 and L(1) = -4. Lastly, let p be the polynomial given by p = 2x +1 – x². (a) Find the matrix representation H = Repg,B(L) of the map h with respect to the basis B. (b) Find the column vector representation, Repâ(p) or [p]g, of the polynomial p with respect to the basis B. (c) Use the matrix obtained in part (a) and the result of part (b) to find the column vector [L(p)]g.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 24EQ

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