When there are two ordered bases for the same vector space, the change of basis matrix from the first basis to the second one, is the matrix that allows us to get the coordinate vector relative to the second basis by using only the coordinate vector relative to the first one, without even knowing the bases themselves. One of the useful features of a basis is that it enables to introduce the concept of coordinates, which are the coefficients of the linear combination when expressing a vector in terms of the vectors of the basis, and they are always specified relative to an ordered basis.Īn important concept related to basis and coordinates is the change of basis matrix. A basis for a vector space V of dimension n is a set of n vectors with the property that every vector in the space can be expressed uniquely as a linear combination of the basis vectors.
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