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San José State University
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applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley U.S.A. |
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The Matrix Representation of Sets of Linear Combinations of Vectors |
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Consider a set of m vectors {X1, …, Xm} and a set of p linear combinations of these vectors {Yi=Σ1mci,jXj: i=1,…,p}.
If Y is the matrix formed by adjoining the p Yi vectors and X is the matrix formed by adjoining the m Xi vectors. Let C denote the p×m matrix of coefficients. This means that ci,j represents the coefficient for Xj in the i-th linear combination. Then
where CT 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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