(c) Find an n such that P₂ is isomorphic to Rn. (Don't just state the value of n; prove why the vector spaces are isomorphic.)
We need the following definition for part (3) below.
(i) φ(u) = φ(v) ⇒ u = v for all u, v∈V; and
(ii) for all w ∈ W there exists v ∈ V such that φ(v)=w.
Consider the set P2 := {f : R → R : f (x) = ax2 + bx + c for some a, b, c ∈ 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
λ(ax2 +bx+c) = λax2 + λbx + λc.
Consider the function D : P2 → P2 given by D(ax2 + bx + c) = 2ax + b.
---------------
c) Find an n such that P2 is isomorphic to Rn. (Don’t just state the value of n; prove why the vector spaces are isomorphic.)
(d) Let φ: P2 →Rn be the function from (c) you used to show that P2 and Rn are isomorphic (where n is your value from (c)). Find a matrix AD such that φ ◦ D ◦ φ−1 is given by left multiplication by AD.
(e) Find the eigenvalues and eigenspaces of AD
![. We need the following definition for part (c) below. Vector spaces V and W are isomorphic
if there is a linear transformation o: V → W that is a bijection, in the sense that
(i) ø(u) = 6(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) = ax? + 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
A(ax? + bx + c) = Xax² + Abx + dc.
Consider the function D: P2 – P2 given by D(ax² + bx + c) = 2ax + b.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12e09b27-396c-49f5-bcab-47e4a186da8a%2F0f785a5c-d1e2-4fb1-bd38-2d0989fdc48e%2Fuf45izd_processed.png&w=3840&q=75)
![(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 : P, → R" be the function from (c) you used to show that P, and R" are
isomorphic (where n is your value from (c)). Find a matrix Ap such that oo 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%2F12e09b27-396c-49f5-bcab-47e4a186da8a%2F0f785a5c-d1e2-4fb1-bd38-2d0989fdc48e%2Fdv20j3_processed.png&w=3840&q=75)
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