The linear combination should be rewritten as WBSince the coordinate mapping x→[x] is a(n)Bmaps to [w], namely the combinationabove.=w whichwhich can be seen in the rewritten expressiontransformation, there existsThis shows the "if" portion of the proof. The "only if" portion can be constructed by applying the same logic inreverse. Given vectors u₁.. up and w in V, show that w is a linear combination of u₁,linear combination of the coordinate vectors [u, 1.....[u].To show the "if" part of the statement, assume that[₂]='41][w] =Which of the following facts can be directly used to rewrite this linear combination?[₂] Bt[w]up if and only if [ [W] Bis a linear combination of the coordinate vectorsWrite this linear combination below.O A. The coordinate mapping x→[x] is a linear transformation.XOB. The vectors u₁.. up can each be written as a linear combination of the vectors in the basis B.O c. The coordinate mapping x→[x] is a one-to-one transformation.OD. The basis B spans V.is a

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is a Given vectors U₁, ..., up and w in V, show that w is a linear combination of u₁,..., up if and only if W linear combination of the coordinate vectorsu, 1 [u]. / * [W]B' To show the "if" part of the statement, assume that w ¹]... .. [ "P]B`` Write this linear combination below. [w] = Which of the following facts can be directly used to rewrite this linear combination? is a linear combination of the coordinate vectors OA. The coordinate mapping x-[x] is a linear transformation. B. The vectors u₁,..., up can each be written as a linear combination of the vectors in the basis B. O c. The coordinate mapping x-[x] is a one-to-one transformation. D. The basis B spans V.