Is the Linear System inconsistent?

[Mr.Tibbs]

Hey guys, I'd assume that this will be a very easy answer. I need to clarify what is an inconsistent linear system. I've worked out my matrix and have achieved this form:

x+0y+z+0w=0
0x+0y+z+w=0
0x+0y+0z+w=0

I know that an inconsistent linear system has no solution i.e 0=1. This is throwing me off because of the y components being all zero which means I can make y what ever I want, but does this also make the system inconsistent?

 

[Dick]

Hey guys, I'd assume that this will be a very easy answer. I need to clarify what is an inconsistent linear system. I've worked out my matrix and have achieved this form:

x+0y+z+0w=0
0x+0y+z+w=0
0x+0y+0z+w=0

I know that an inconsistent linear system has no solution i.e 0=1. This is throwing me off because of the y components being all zero which means I can make y what ever I want, but does this also make the system inconsistent?

x=y=z=w=0 is a solution. It will always be a solution when the RHS is all zeros. You have an infinite number of solutions, but it's not inconsistent. Inconsistent means no solutions.

 

[Mr.Tibbs]

Awesome! Thanks for clearing that up for me!

 

[Ray Vickson]

Hey guys, I'd assume that this will be a very easy answer. I need to clarify what is an inconsistent linear system. I've worked out my matrix and have achieved this form:

x+0y+z+0w=0
0x+0y+z+w=0
0x+0y+0z+w=0

I know that an inconsistent linear system has no solution i.e 0=1. This is throwing me off because of the y components being all zero which means I can make y what ever I want, but does this also make the system inconsistent?

"Inconsistent" does not apply to homogeneous linear systems (where the right-hand-sides are all zero) because setting all variables = 0 *does* give a solution. No: "consistency" and its opposite are relevant when (some of the) right-hand sides are *nonzero*. For example, the system x = 0 and 2*x = 0 is perfectly consistent, but the system x = 1 and 2*x = 3 is inconsistent.