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Change of basis/R^2/Standard and 12,-23/Example

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We consider in the standard basis,

and the basis

The basis vectors of can be expressed directly with the standard basis, namely

Therefore, we get immediately

For example, the vector which has with respect to the coordinates , has the coordinates

with respect to the standard basis . The transformation matrix is more difficult to compute: We have to write the standard vectors as linear combinations of and . A direct computation (solving two linear systems) yields

and

Hence,