anonymous
  • anonymous
"A system in echelon form can be inconsistent." Can someone give me an explanation as to why this is false?
Mathematics
  • Stacey Warren - Expert brainly.com
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chestercat
  • chestercat
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dan815
  • dan815
any system in echelon form can be rewritten in row reduced echelon form
dan815
  • dan815
|dw:1439931493101:dw|
dan815
  • dan815
so u cannot have inconsistency

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anonymous
  • anonymous
Isn't a matrix in echelon form if it has a row of zeroes on the bottom? Like, is the following not in echelon form? \[\begin{bmatrix}1&&1&&1\\0&&0&&1\end{bmatrix}\](The system in this case would be \(x + y = 1\) and \(0x + 0y = 1\))
anonymous
  • anonymous
I just didn't see anything in my book's definition of echelon form that prevented this, but I'm probably wrong on that.
dan815
  • dan815
that has 3 variables x ,y ,z we dont know the constants
dan815
  • dan815
|dw:1439931839159:dw|
anonymous
  • anonymous
I meant for there to to be the line for an augmented matrix between columns 2 and 3, but I don't know how to do that in \(\LaTeX\) just yet.
dan815
  • dan815
|dw:1439931879525:dw|
dan815
  • dan815
this is not echelon form there is simply no echelon form for this
dan815
  • dan815
http://prntscr.com/866zua
dan815
  • dan815
actually wait i dont know, what does inconsistent mean again?
dan815
  • dan815
there is actually an example there where RREF has all 0s in the bottom soo
anonymous
  • anonymous
Yeah, that's what's confusing me.
dan815
  • dan815
i was thining inconsistent is same as dependant but thats not it
anonymous
  • anonymous
They said \[\begin{bmatrix}1~~~1~~~3\\0~~~0~~~2\end{bmatrix}\]isn't in ehcelon form but \[\begin{bmatrix}1~~~1~~~3\\0~~~0~~~1\end{bmatrix}\]is?\[\\\]And my book cites inconsistent as having no solution.
dan815
  • dan815
gotta have it as 1 i guess
anonymous
  • anonymous
Hmm...but if the second one is allowed, then is it assumed that the matrix has 3 unknowns, but we don't know the constant matrix since there is no divider?
dan815
  • dan815
what is the definition of inconsistent, that should clear this up
anonymous
  • anonymous
My book's definition of inconsistent is that there is no solution that satisfies all parts of the system simultaneously. "Inconsistent: No pair of numbers (s1,s2) satisfies all three equations simultaneously."
anonymous
  • anonymous
And to clarify, the question on my assignment was asking if the above statement was true or false. The answer ended up being false, and I just can't seem to figure out why.
dan815
  • dan815
Ohh okay
anonymous
  • anonymous
Is it just something weird based on definition or am I way off? Also, here's my book's definition of echelon form if it helps. "In each row of a system, the first variable with a nonzero coefficient is the row’s leading variable. A system is in echelon form if each leading variable is to the right of the leading variable in the row above it, except for the leading variable in the first row, and any all-zero rows are at the bottom."
anonymous
  • anonymous
They use those examples to show you how a matrix is called echelon. \(\left[\begin{matrix}1&1&3\\0&0&2\end{matrix}\right]\) is not an echelon matrix because the leading 1 of the second row is 2, not 1. But you can get the echelon form by divided row2 by 2 to get \(\left[\begin{matrix}1&1&3\\0&0&1\end{matrix}\right]\)
anonymous
  • anonymous
That is just NOTATION.
anonymous
  • anonymous
That shows you how "an echelon form can be inconsistent"
anonymous
  • anonymous
Ok, I'm pretty sure the leading one's thing is just differences in definition between texts (I cross checked multiple books, and some said they HAD to have leading one's and others didn't care.)
anonymous
  • anonymous
But the question answer was that the statement was false, indicating that it cannot be inconsistent...That's what I was confused on.
anonymous
  • anonymous
It's above my head!! :)
anonymous
  • anonymous
lol ok. I'm just as confused as you are, and this is the assignment from my first Linear Algebra class XD
anonymous
  • anonymous
I have to go, but @ganeshie8 , if you could take a look at this whenever you get online, that'd be great!
ganeshie8
  • ganeshie8
The question is about "system of equations", not about a "matrix" being inconsistent. We say a "system of equations" is consistent or inconsistent. But never say a matrix is inconsistent or inconsistent. If the system of equations is inconsistent, then after eliminations, the augmented matrix in echelon form will have at least one row in the bottom with all 0's except for the last entry. For example : \(\begin{bmatrix}1&2&5&|&5\\0&0&0&|&\color{red}{3}\end{bmatrix}\).
ganeshie8
  • ganeshie8
The given statement, "A system in echelon form can be inconsistent." means that the "starting" system of equations itself are in echelon form, for example : \[ \begin{array}{} x&+&3y&+&z&=&5\\ &&y&+&5z&=&3\\ &&&&z&=&8 \end{array}\] In other words, the given system of equations itself is in some kind of triangular/staircase form. Our task is to prove that any system like this can never be inconsistent..
anonymous
  • anonymous
So, when it said "in echelon form", it meant that the system was given in echelon form, not manipulated into it?
ganeshie8
  • ganeshie8
Exactly! the "given" set of equations are in "staircase" looking form
ganeshie8
  • ganeshie8
The given statement is false because the system of equations in that staircase form "always has a solution" by back substitution.
ganeshie8
  • ganeshie8
heard of "back substitution" before ?
anonymous
  • anonymous
Oh! That makes a lot more sense. I guess it was just the wording that I misinterpreted. And yes, I know what back substitution is. I've self-studied Linear Algebra before, but I never formally studied it in a classroom setting (which I am now doing).
ganeshie8
  • ganeshie8
Fact : you can never learn linear algebra fully, it is a vast subject and there is always something we never thought of..
anonymous
  • anonymous
haha seems like it. My teacher started off lecture stressing a similar point, bringing in lots of different fields where it is applied. Anyway, thanks so much for the help. :)

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