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].

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Problem 1. Consider the vector space P2 [cc,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[sc,;y] = {a0 +a11L' +a2y + :1ng + a4y2 +a5$y I 031, - . .,a5 6 R} Consider the subset Paymh, y] of P2 [53, 9] consisting of polynomials 30(13, y) satisfying that 33(33 0!) = p(d,c) for all c, d E R, i.e. the polynomials which doesn't change when we interchange cc and y. a) Show that ngmm 9] is a subspace of P2 [13, y] b) Find the dimension of Paymkc, y]