By Prabhat Choudhary

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Note, that the process cannot continue infinitely, because a linearly independent system of vectors in V cannot have more than n = dim V vectors. , ofthe solution set) of a linear system. , a homogeneous system is a system of form Ax = O. With each system Ax = b we can associate a homogeneous system just by putting b = O. Theorem. (General solution of a linear equation). Let a vector xlsatisfy the equation Ax = b, and let H be the set of all solutions of the associated homogeneous system Ax = O.

1 = 12 + 3 -10 + 2 = 7. Consider next q31. 3 =-1 +0+0+ 18= 17. Consider finally q32. 1 = - 4 + 0 + (- 14) + 6 = - 12. We therefore conclude that AB=[ ~ 41 53 2-Ij o -1 7 6 1 4 2 3 0 -2 3 +~ 17 Example. Consider again the matrices A~P -1 4 3 1 5 0 7 -Ij ~ and B= 1 4 2 3 0 -2 3 137 1. -12 56 Matries Note that B is a 4 x 2 matrix and A is a 3 x 4 matrix, so that we do not have a definition for the "product" BA. We leave the proofs of the following results as exercises for the interested reader. Proposition.

Proposition. Any basis in IR n must have exactly n vectors in it. Proof This fact foIlows immediately from the previous proposition, but there is also a direct proof. •• , vm be a basis in IR n and let A be the n x m matrix with columns VI' v2' ... , vm . The fact that the system is a basis, means that the equation Ax = b has a unique solution for any (all possible) right side b. The existence means that there is a pivot in every row (of a reduced echelon form of the matrix), hence the number of pivots is exactly n.