Problem 1. 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] = {ao +a1x + a2Y+ azx² + a4y´ + a5xy | a1, . ., a5 ER} sym Consider the subset PYm[x, y] of P2[x, y] consisting of polynomials p(x, y) satisfying that p(c, d) = p(d, c) for all c, d e R, i.e. the polynomials which doesn't change when we interchange x and y. a) Show that P™ [x, y] is a subspace of P2[x, y]. sym b) Find the dimension of P" [x, y].
Problem 1. 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] = {ao +a1x + a2Y+ azx² + a4y´ + a5xy | a1, . ., a5 ER} sym Consider the subset PYm[x, y] of P2[x, y] consisting of polynomials p(x, y) satisfying that p(c, d) = p(d, c) for all c, d e R, i.e. the polynomials which doesn't change when we interchange x and y. a) Show that P™ [x, y] is a subspace of P2[x, y]. sym b) Find the dimension of P" [x, y].
Problem 1. 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] = {ao +a1x + a2Y+ azx² + a4y´ + a5xy | a1, . ., a5 ER} sym Consider the subset PYm[x, y] of P2[x, y] consisting of polynomials p(x, y) satisfying that p(c, d) = p(d, c) for all c, d e R, i.e. the polynomials which doesn't change when we interchange x and y. a) Show that P™ [x, y] is a subspace of P2[x, y]. sym b) Find the dimension of P" [x, y].
Linear algebra question. Please explain how to answer this.
Transcribed Image Text:**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 \(\mathbb{P}_2^{\text{sym}}[x, y]\).
Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.
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![**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 \(\mathbb{P}_2^{\text{sym}}[x, y]\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbcd30ec2-5467-4479-86f4-f6a2b30f1318%2Fbbf806b4-2299-4c93-b2ae-db079dab7033%2Fckh1fd9_processed.png&w=3840&q=75)
