Consider the vector space P2[x, y](no need to show that it is a vector space) of polynomials in two variables of total degree less than or equal to 2 (with the usual addition and scalar multiplication of polynomials), i.e. P2[x, y] = {a0 + a1x + a2y + a3x 2 + a4y 2 + a5xy | a1, . . . , a5 ∈ R} Consider the subset P sym 2 [x, y] of P2[x, y] consisting of polynomials p(x, y) satisfying that p(c, d) = p(d, c) for all c, d ∈ R, i.e. the polynomials which doesn't change when we interchange x and y.a) Show that P sym 2 [x, y] is a subspace of P2[x, y].b) Find the dimension of P sym 2 [x, y].please explain in detail. photo below **Problem 1**Consider the vector space $ \mathbb{P}_2[x, y] $ (no need to show that it is a vector space) of polynomials in two variables of total degree less than or equal to 2 (with the usual addition and scalar multiplication of polynomials), i.e.\[\mathbb{P}_2[x, y] = \{a_0 + a_1 x + a_2 y + a_3 x^2 + a_4 y^2 + a_5 xy \mid a_1, \ldots, a_5 \in \mathbb{R}\}\]Consider the subset $ \mathbb{P}_2^{\text{sym}}[x, y] $ of $ \mathbb{P}_2[x, y] $ consisting of polynomials $ p(x, y) $ satisfying that $ p(c, d) = p(d, c) $ for all $ c, d \in \mathbb{R} $, i.e. the polynomials which don’t change when we interchange $ x $ and $ y $.a) Show that $ \mathbb{P}_2^{\text{sym}}[x, y] $ is a subspace of $ \mathbb{P}_2[x, y] $.b) Find the dimension of $ \mathbb{P}_2^{\text{sym}}[x, y] $.

Consider the vector space P2[x, y](no need to show that it is a vector space) of polynomials in two variables of total degree less than or equal to 2 (with the usual addition and scalar multiplication of polynomials), i.e. ℙ2x, y=a0+a1x+a2y+a3x2+a4y2+a5xy a0, a1, ... , a5∈ℝ Now, consider the subset ℙ2symx, y of ℙ2x, y consisting of polynomials p(x, y) satisfying that p(c, d)=p(d, c) for all c, d∈ℝ, i.e. the polynomials which don't change when we interchange x and y. For (a), We need to show that ℙ2symx, y is a subspace of ℙ2x, y. Let p(x, y), q(x, y)∈ℙ2x, y and α, β∈ℝ. Now, αp(x, y)+βq(x, y)=αa0+a1x+a2y+a3x2+a4y2+a5xy+βb0+b1x+b2y+b3x2+b4y2+b5xy=αa0+βb0+αa1+βb1x+αa2+βb2y+αa3+βb3x2 +αa4+βb4y2+αa5+βb5xy∈ℙ2x, y Thus, ℙ2symx, y is a subspace of ℙ2x, y.For (b), We need to find the dimension of ℙ2symx, y. As per definition, if p(x,y)∈ℙ2symx, y, then p(x, y)=p(y, x). So, p(x, y)=p(y, x)⇒a0+a1x+a2y+a3x2+a4y2+a5xy=a0+a1y+a2x+a3y2+a4x2+a5yx⇒ a1x+a2y+a3x2+a4y2=a1y+a2x+a3y2+a4x2After equating the coefficients we get, ⇒ a1=a2, a3=a4 Thus, ℙ2symx, y=a0+a1x+a1y+a3x2+a3y2+a5xy a0, ... , a5∈ℝ =a0+a1x+y+a3x2+y2+a5xy a0, a1, a3, a5∈ℝ Answer: Hence, the dimension of ℙ2symx, y is 4. And the basis is 1, x+y, x2+y2, xy.

Answered: Problem 1. Consider the vector space…

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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Consider the vector space P2[x, y](no need to show that it is a vector space) of polynomials in two variables of total degree less than or equal to 2 (with the usual addition and scalar multiplication of polynomials), i.e. P2[x, y] = {a0 + a1x + a2y + a3x 2 + a4y 2 + a5xy | a1, . . . , a5 ∈ R} Consider the subset P sym 2 [x, y] of P2[x, y] consisting of polynomials p(x, y) satisfying that p(c, d) = p(d, c) for all c, d ∈ R, i.e. the polynomials which doesn't change when we interchange x and y.

a) Show that P sym 2 [x, y] is a subspace of P2[x, y].

b) Find the dimension of P sym 2 [x, y].

please explain in detail. photo below

**Problem 1**

Consider the vector space $ \mathbb{P}_2[x, y] $ (no need to show that it is a vector space) of polynomials in two variables of total degree less than or equal to 2 (with the usual addition and scalar multiplication of polynomials), i.e.

\[
\mathbb{P}_2[x, y] = \{a_0 + a_1 x + a_2 y + a_3 x^2 + a_4 y^2 + a_5 xy \mid a_1, \ldots, a_5 \in \mathbb{R}\}
\]

Consider the subset $ \mathbb{P}_2^{\text{sym}}[x, y] $ of $ \mathbb{P}_2[x, y] $ consisting of polynomials $ p(x, y) $ satisfying that $ p(c, d) = p(d, c) $ for all $ c, d \in \mathbb{R} $, i.e. the polynomials which don’t change when we interchange $ x $ and $ y $.

a) Show that $ \mathbb{P}_2^{\text{sym}}[x, y] $ is a subspace of $ \mathbb{P}_2[x, y] $.

b) Find the dimension of $ \mathbb{P}_2^{\text{sym}}[x, y] $.

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