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Proofs from Coordinates and bases

All proofs from Coordinates and bases.

[PROOF]

Write $v=x_1u_1+\cdots+x_nu_n$ . Taking the dot product with $u_k$ gives

u_k\cdot v =\sum_{j=1}^n x_j(u_k\cdot u_j) =x_k.

These are exactly the entries of $[v]_B$ .

Part of the thread Coordinates and linear structure, in 5 places.

A basis turns vectors and maps into columns and matrices; a good basis makes the calculation simple.

DFCoordinate vectorcoordinates in a basisALG 2 / Here