I have the solutions to thesequestions but they are very brief and I do notunder

Question

I have the solutions to thesequestions but they are very brief and I do notunderstand why they are the answers. I was hoping someone couldclear them up a bit. The answers are provided in boldbelow. Your help isappreciated. A system of eight linearequations in eight unknowns never has a unique solution, regardlessof the constants on the right sides of the equations.
(a) Will the system always be consistent, regardless of theconstants on the right sides of the equations? Justify youranswer. No (b) What can you say about thedimension of the null space of the coefficient matrix
of this system? Justify your answer. dim of Nul A is atleast 1 I have the solutions to thesequestions but they are very brief and I do notunderstand why they are the answers. I was hoping someone couldclear them up a bit. The answers are provided in boldbelow. Your help isappreciated. A system of eight linearequations in eight unknowns never has a unique solution, regardlessof the constants on the right sides of the equations.
(a) Will the system always be consistent, regardless of theconstants on the right sides of the equations? Justify youranswer. No (b) What can you say about thedimension of the null space of the coefficient matrix
of this system? Justify your answer. dim of Nul A is atleast 1

Explanation / Answer

x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 = 2
2x1 + 2x2 + 2x3 + 2x4 + 2x5 + 2x6 + 2x7 + 2x8 = 5
Row-reducing yields 0 = 1, which is false.
b) The key here is that vectors in the null space are writtenin terms of their free variables. For example.
x1 + x3 = 0 x2 = 0 0 = 0
x1 = -x3 No relation between x2 and x3 x3 = x3
So the null space written with respect to x3 is [-1, 0,1].
Given that the 8x8 matrix is not unique (meaning at least 1free variable), there will be at least 1 vector in the nullspace.
Thus dim(NS(A))>=1

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