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Consider a matrix A that represents the augmented matrix of a linear system in 4 variable: w, x, y, and z. rref(A) is:

[ 0 1 0 0 2
  0 0 1 0 3
  0 0 0 1 5 ]

What does it mean for the first column of rref(A) to be all 0 (zero).

Does this mean that w can be anything. Would this then also mean that the solution to the system would be a straight line (infinitely many solutions).

Thank you for the help.

Ben Grossmann
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KingRand16
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  • Assuming no contradictions in the system (i.e on the left side there is 0, and on the right side, not 0) then yes, there will be infinitely many solutions to these equations. This is known as the degree of freedom. – Eminem Jan 24 '21 at 21:48
  • Remember, that matrix corresponds to a system of equations. In your example, the equations, in the four unknowns $w,x,y,z$, are $x=2$; $y=3$; $z=5$. So the general solution is $(w,x,y,z)=(t,2,3,5)$, where $t$ is arbitrary. Infinitely many solutions, all in a straight line (in four dimensions). – Gerry Myerson Jan 24 '21 at 22:30
  • Have these comments been helpful, King? – Gerry Myerson Jan 26 '21 at 04:14
  • It's not polite to ask for help and then ignore the people who try to help you, King. – Gerry Myerson Jan 27 '21 at 12:20

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