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242LN7.3

# 242LN7.3 - 7.3 Coordinates and Change of Basis If B =...

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7.3 Coordinates and Change of Basis If B = { ~v 1 ,~v 2 , . . . ,~v k } is a basis for a subspace W , then every vector ~u in W can be written in the form c 1 ~v 1 + c 2 ~v 2 + · · · + c k ~v k because B spans W . At the same time, we cannot have two different linear combinations of the ~v i ’s that add up to ~u , because we could subtract one from the other and get a non-trivial linear combination of the ~v i ’s that adds up to ~ 0 ; this is because B is linearly independent. So this representation of ~u is unique!

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If we arrange the vectors in a basis in a certain order, we can put these numbers c i into a vector. This vector of the c i ’s is called the of ~u with respect to the ordered basis B and is sometimes denoted [ ~u ] B . When the order of the vectors in a basis matters, the basis is often written using parentheses: ( ~v 1 ,~v 2 , . . . ,~v k ) . This is called an .
Finding the coordinates of a vector ~u with respect to a basis B is just like determining whether ~u is in the span of B , except that there will be exactly one solution when it is! So we can re-use our procedure

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242LN7.3 - 7.3 Coordinates and Change of Basis If B =...

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