(d) Let ø : P2 → R" be the function from (c) you used to show that P2 and R" are isomorphic (where n is your value from (c)). Find a matrix Ap such that po Do0-1 is given by left multiplication by Ap.
(d) Let ø : P2 → R" be the function from (c) you used to show that P2 and R" are isomorphic (where n is your value from (c)). Find a matrix Ap such that po Do0-1 is given by left multiplication by Ap.
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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Please help me with part d
![We need the following definition for part (c) below. Vector spaces V and W are isomorphic
if there is a linear transformation p: V → W that is a bijection, in the sense that
(i) ø(u) = $(v)
u = v for all u, v e V; and
(ii) for all w E W there exists v e V such that o(v)
= W.
Consider the set P2 := {f: R → R : f(x) = ax² + bx + c for some a, b, c E R}. So P2 is
the set of polynomials of degree up to 2. This set is a vector space over R under addition
given by
(a1x2 + b1x + c1) + (a2x² + b2x + c2) = (a1 + a2)x² + (b1 + b2)x + (c1 + c2),
and scalar multiplication given by
X(ax? + bx + c) = Aax² + Abx + dc.
Consider the function D: P2 –→ P2 given by D(ax² + bx + c) = 2ax + b.
(a) Why did I call this function D?
(b) Prove that D is a linear transformation'.
(c) Find an n such that P2 is isomorphic to R". (Don't just state the value of n; prove
why the vector spaces are isomorphic.)
(d) Let ø : P2 → R" be the function from (c) you used to show that P2 and R" are
isomorphic (where n is your value from (c)). Find a matrix Ap such that po Doo-1
is given by left multiplication by Ap.
(e) Find the eigenvalues and eigenspaces of Ap.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9ee89080-ea45-47d2-be13-9cc64e04f9b1%2Fb0176aa4-9bc3-488b-8d3f-89357e3c7640%2F4q0dbxi_processed.png&w=3840&q=75)
Transcribed Image Text:We need the following definition for part (c) below. Vector spaces V and W are isomorphic
if there is a linear transformation p: V → W that is a bijection, in the sense that
(i) ø(u) = $(v)
u = v for all u, v e V; and
(ii) for all w E W there exists v e V such that o(v)
= W.
Consider the set P2 := {f: R → R : f(x) = ax² + bx + c for some a, b, c E R}. So P2 is
the set of polynomials of degree up to 2. This set is a vector space over R under addition
given by
(a1x2 + b1x + c1) + (a2x² + b2x + c2) = (a1 + a2)x² + (b1 + b2)x + (c1 + c2),
and scalar multiplication given by
X(ax? + bx + c) = Aax² + Abx + dc.
Consider the function D: P2 –→ P2 given by D(ax² + bx + c) = 2ax + b.
(a) Why did I call this function D?
(b) Prove that D is a linear transformation'.
(c) Find an n such that P2 is isomorphic to R". (Don't just state the value of n; prove
why the vector spaces are isomorphic.)
(d) Let ø : P2 → R" be the function from (c) you used to show that P2 and R" are
isomorphic (where n is your value from (c)). Find a matrix Ap such that po Doo-1
is given by left multiplication by Ap.
(e) Find the eigenvalues and eigenspaces of Ap.
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