# An homogeneous system is always consistent theorem

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Unformatted text preview: An homogeneous system is always consistent (Theorem HSC ) and with n = 3, r = 2 an application of Theorem FVCS  yields one free variable. With a suitable choice of x 3 each solution can be written in the form -x 3-x 3 x 3 C30 Contributed by Robert Beezer Statement  Deﬁnition NSM  tells us that the null space of A is the solution set to the homogeneous system LS ( A, ). The augmented matrix of this system is 2 4 1 3 8 0-1-2-1-1 1 0 2 4-3 4 0 2 4-1-7 4 0 To solve the system, we row-reduce the augmented matrix and obtain, 1 2 5 1-8 0 1 2 Version 2.12 Subsection HSE.SOL Solutions 77 This matrix represents a system with equations having three dependent variables ( x 1 , x 3 , and x 4 ) and two independent variables ( x 2 and x 5 ). These equations rearrange to x 1 =-2 x 2-5 x 5 x 3 = 8 x 5 x 4 =-2 x 5 So we can write the solution set (which is the requested null space) as N ( A ) = -2 x 2-5 x 5 x 2 8 x 5-2 x 5 x 5 | x 2 ,x 5 ∈ C C31 Contributed by Robert Beezer Statement  We form the augmented matrix of the homogeneous system LS ( B, ) and row-reduce the matrix, -6 4-36 6 2-1 10-1 0-3 2-18 3 RREF----→ 1 2 1 0 1-6 3 0 0 0 We knew ahead of time that this system would be consistent (Theorem HSC ), but we can now see there are n-r = 4-2 = 2 free variables, namely x 3 and x 4 (Theorem FVCS ). Based on this analysis, we can rearrange the equations associated with each nonzero row of the reduced row-echelon form into an expression for the lone dependent variable as a function of the free variables. We arrive at the solution set to the homogeneous system, which is the null space of the matrix by Deﬁnition NSM , N ( B ) = -2 x 3-x 4 6 x 3-3 x 4 x 3 x 4 | x 3 , x 4 ∈ C M50 Contributed by Robert Beezer Statement  Since the system is homogeneous, we know it has the trivial solution (Theorem HSC ). We cannot say anymore based on the information provided, except to say that there is either a unique solution or inﬁnitely many solutions (Theorem PSSLS ). See Archetype A  and Archetype B  to understand the possibilities. M51 Contributed by Robert Beezer Statement  Since there are more variables than equations, Theorem HMVEI  applies and tells us that the solution set is inﬁnite. From the proof of Theorem HSC  we know that the zero vector is one solution. M52 Contributed by Robert Beezer Statement  By Theorem HSC , we know the system is consistent because the zero vector is always a solution of a homogeneous system. There is no more that we can say, since both a unique solution and inﬁnitely many solutions are possibilities....
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