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

[negation]

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

Homework Statement

"For every b ∈ R7, the system A^Tx = 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?

 

[HallsofIvy]

Are you told what A is or is it a general matrix?

 

[negation]

Are you told what A is or is it a general matrix?

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.

 

[Ray Vickson]

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.

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
$$x_1 + x_2 = 1\\ 3x_1 + 3x_2 = 4$$