Concern a vector space V, a basis B b 1 , ,b n ), and the coordinate mapping x x B Given vectors u 1 , ,u p , and w in V, show that w is a linear combination of u 1 , ,u p if and only if w B is a linear combination of the coordinate vectors u 1 B , , u p B

Question: Concern a vector space V, a basis B = {b 1 ,...,b n ), and the coordinate mapping x [x] B . Given vectors

Concern a vector space V, a basis B = {b₁,...,b_n), and the coordinate mapping x ↦ [x] _B.

Given vectors u₁,...,u_p, and w in V, show that w is a linear combination of u₁,...,u_p if and only if [w]_B is a linear combination of the coordinate vectors [u₁]_B,...,[u_p]_B.

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