Linear algebra question. Please explain how to answer this. **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.,\[\mathbb{P}_2[x, y] = \{a_0 + a_1x + a_2y + a_3x^2 + a_4y^2 + a_5xy \mid a_1, \ldots, a_5 \in \mathbb{R}\}\]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 \(\mathbb{P}_2^{\text{sym}}[x, y]\).

Given ℙ2x,y is a vector space of polynomials in two variables of total degree less than or equal to 2, i.e., ℙ2x,y=a0+a1x+a2y+a3x2+a4y2+a5xy| a0, a1, a2, a3, a4, a5∈ℝ Consider the subset ℙ2symx,y of ℙ2x,y consisting of polynomials p(x,y) satisfying that p(c,d)=p(d,c) ∀c,d∈ℝ (a) Consider the zero vector θ(x,y)=0 ∀x,y θ(x,y)=0=θ(y,x) ∀ x,y ∈ℝ Thus, θ(x,y)∈ℙ2symx,y ∀x,y ∴ℙ2symx,y≠∅Next, let p(c,d), q(c,d) ∈ℙ2symx,y Let W(c,d)=p(c,d)+q(c,d) Then, W(c,d)=p(c,d)+q(c,d)=p(d,c)+q(d,c) ∵(p(c,d), q(c,d) ∈ℙ2symx,y=W(d,c) ∀c,d Thus, W(c,d) ∈ℙ2symx,y Next, let p(c,d)∈ℙ2sym and let r∈ℝ Then, rp(c,d)=rp(c,d)=rp(d,c) ∵p(c,d)∈ℙ2symx,y=rp(d,c) Thus, rp(c,d) ∈ℙ2symx,y Hence, ℙ2symx,y is a subspace of ℙ2x,y(b) Note that any element p(c,d) ∈ℙ2symx,y will be of the form, p(c,d)=a0+a1c+a2d+a3c2+a4d2+a5cd ⋯⋯(1) Now, p(c,d)=p(d,c)a0+a1c+a2d+a3c2+a4d2+a5cd=a0+a1d+a2c+a3d2+a4c2+a5dc This gives, a1=a2a3=a4 Then, from (1), p(c,d)=a0+a1c+a1d+a3c2+a3d2+a5dc=a0+a1c+d+a3c2+d2+a5dcThus, every element in ℙ2symx,y is a linear combination of the polynomials 1, c+d, c2+d2,cd. In other words, W⊆span1, c+d, c2+d2,cd Let S=1, c+d, c2+d2,cd Note that the set S is linearly independent. The dimension of ℙ2symx,y is given by S=4. Thus, dimℙ2symx,y=4