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The Matrix Representation of Sets of Linear Combinations of Vectors 

Consider a set of m vectors {X_{1}, …, X_{m}} and a set of p linear combinations of these vectors {Y_{i}=Σ_{1}^{m}c_{i,j}X_{j}: i=1,…,p}.
If Y is the matrix formed by adjoining the p Y_{i} vectors and X is the matrix formed by adjoining the m X_{i} vectors. Let C denote the p×m matrix of coefficients. This means that c_{i,j} represents the coefficient for X_{j} in the ith linear combination. Then
where C^{T} is the transpose of the coefficient matrix C. The number of rows of C must be m and the number of columns of C must be the same as the number of columns of Y; i.e. p. The matrix C could have been defined the other way around so that tranposition was not involved, but the definition of C seemed to be the natural way of defining C.
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