Let Pn denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: P3 → P2 be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) P3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + α₁x + αº and q(x) = b3x³ + b₂x² + b₁x + b₁ be any two polynomials in P3 and c E R. a. D(p(x) + q(x)) = . (Enter a3 as a3, etc.) D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = P3? choose b. D(cp(x)) = + c(D(p(x))) = î Does D(cp(x)) = c(D(p(x))) for all CER and all p(x) = P3? choose ◆ c. Is D a linear transformation? choose ◆
Let Pn denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: P3 → P2 be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) P3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + α₁x + αº and q(x) = b3x³ + b₂x² + b₁x + b₁ be any two polynomials in P3 and c E R. a. D(p(x) + q(x)) = . (Enter a3 as a3, etc.) D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = P3? choose b. D(cp(x)) = + c(D(p(x))) = î Does D(cp(x)) = c(D(p(x))) for all CER and all p(x) = P3? choose ◆ c. Is D a linear transformation? choose ◆
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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Transcribed Image Text:Let Pn denote the vector space of
polynomials in the variable x of degree n or less
with real coefficients. Let D : P3 - P2 be the
function that sends a polynomial to its derivative.
That is, D(p(x)) = p' (x) for all polynomials
p(x) = P3. Is D a linear transformation?
Let p(x)
=
a³x³ + ²x² + a₁ + a₁ and
q(x) = b3x³ + b₂x² + b₁x + b₁ be any two
polynomials in P3 and c E R.
a. D(p(x) + q(x)) =
. (Enter a3 as a3, etc.)
D(p(x)) + D(q(x)) =
Does
D(p(x) + q(x)) = D(p(x)) + D(q(x))
for all p(x), q(x) = P3?
choose
b. D(cp(x)) =
+
c(D(p(x))) =
=
Does D(cp(x)) = c(D(p(x))) for all
CER and all p(x) = P3?
choose
✪
c. Is D a linear transformation?
choose
◆
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