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Example of For every b ∈ R7, the system ATx = b is consistent

  1. Apr 3, 2014 #1
    example of "For every b ∈ R7, the system ATx = b is consistent"

    1. The problem statement, all variables and given/known data

    "For every b ∈ R7, the system ATx = b is consistent"


    I'm not sure if this is the right place to post this question. There's isn't a subsection known as 'general math' for me to post.

    What does the above statement implies? Any examples?
     
  2. jcsd
  3. Apr 3, 2014 #2

    HallsofIvy

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    Are you told what A is or is it a general matrix?
     
  4. Apr 3, 2014 #3

    You are given that A is an 8 × 11 matrix of nullity 7.

    There are a few intermediate question before the question in OP. I manage to get the questions correct so there's no issue.
    As for the question in the OP, I suppose I do not understand what the question is implying.
     
  5. Apr 3, 2014 #4

    Ray Vickson

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    Is asks for conditions under which the equations ##A^T x = b## have at least one solution for every possible right-hand-side ##b \in \mathbb{R}^7##.

    However, are you sure you have stated the question correctly? If ##A## is an ##8 \times 11## matrix, ##x## must be in ##\mathbb{R}^{8}## and ##A^T x## is in ##\mathbb{R}^{11}##. Therefore, it would be impossible for a vector in ##b \in \mathbb{R}^7## to be equal to ##A^Tx \in \mathbb{R}^{11}##, no matter how you select ##x##.

    Anyway, in general for a system of equations, "consistency" means that the equations do not ask for the impossible---that is, that the system has at least one solution. "Inconsistency" means the opposite: the system has no solutions at all. An example of an inconsistent system would be
    [tex]x_1 + x_2 = 1\\
    3x_1 + 3x_2 = 4[/tex]
     
    Last edited: Apr 3, 2014
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