An homogeneous system is always consistent theorem

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Unformatted text preview: An homogeneous system is always consistent (Theorem HSC [67]) and with n = 3, r = 2 an application of Theorem FVCS [57] 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 [73] Definition NSM [69] 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 [73] 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 [67]), but we can now see there are n-r = 4-2 = 2 free variables, namely x 3 and x 4 (Theorem FVCS [57]). 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 [69], 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 [74] Since the system is homogeneous, we know it has the trivial solution (Theorem HSC [67]). 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 [57]). See Archetype A [750] and Archetype B [755] to understand the possibilities. M51 Contributed by Robert Beezer Statement [74] Since there are more variables than equations, Theorem HMVEI [69] applies and tells us that the solution set is infinite. From the proof of Theorem HSC [67] we know that the zero vector is one solution. M52 Contributed by Robert Beezer Statement [74] By Theorem HSC [67], 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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