Question

Given the vector space R[2]deg<s of polynomials with real coefficients of degree at most 5, and Ui = {p(z) : p(z) a? + bz5, for abe R}, find a subspace U2 such that R deg< 5 = Ui φ Ủy Is this U2 unique? (g) If V be a finite dimensional vector space, dim V = n and B = 〈ui,u2, . . . , un) is a basis of V, then show that:

Answer 1

Given that: the v be a finite dimensional vector space, dim v = n and B = (u_1, u_2, --- u_n) is a basis of v. the basis of v: the (u_1,--- u_n) spans the n - dimensional vector space v and u_1 + -- + u_n = 0 if any vector u_1 is deleted then the resulting set (u_1, -- u_i + 1, ---, u_n) is a basis for v. step: 1 let A = (u_1, --- u_n) with out loss of generality we will remove u_1 from A. from hypothesis, we have u_1 + -- u_n = 0 \r\n this implies u_i --- (u_1 + --- + u_i + 1 -- u_n) so u_i is the linear combination of rest of the elements in A. let B = (u_1, --- u_i + 1, --- u_n) Because u_1 is a linear space of elements form B and A is span of v. then span of A = span of B = v step: 2 B spans v and cardinality B is n. Also we know that dimension v is n.