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My book has an example where they ask us to find whether a system of linear equations is consistent.

A paraphrased version of the example reads as follows:

The rank of the coefficient matrix is equal to the rank of the augmented matrix.

Original augmented matrix: \begin{bmatrix}1&1&-1&-1\\1&0&1&3\\3&2&-1&1\end{bmatrix} Reduced matrix: \begin{bmatrix}1&0&1&3\\0&1&-2&-4\\0&0&0&0\end{bmatrix}

As shown above, b is in the column space of A, so the system of linear >equations is consistent.

I'm just confused on how they knew that b is in the column space of A. When they say "as shown above", what are they referring to? Does it have something to do with the rank that they mentioned earlier in the example?

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    The system $Ax=b$ is consistent if and only if $b$ can be written as a linear combination of the columns of $A$. Said differently,if and only $b$ is in the column space of $A$. Since the augmented system is consistent (no pivots in the last column), hence $Ax=b$ has a solution and so... – 2017-01-05
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    How do you know that having no pivots in the last column means the augmented system is consistent? – 2017-01-05
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    @dagny If have a row with all 0s except for a non-zero value in the rightmost position, then this corresponds to the equation $0x_1 + 0x_2 + \cdots + 0x_n = b$, which is inconsistent (has no solution). The whole point of row echelon form is that it simplifies the system of equations to such an extent that if there are any inconsistencies then they are always manifested in this obvious form. Since the reduced augmented matrix here has no such inconsistencies, it is consistent (there is no obstacle preventing you from solving for $x_n, x_{n-1}, x_{n-2}, \ldots, x_1$). – 2017-01-05
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    @ErickWong I get why it's consistent, we learned about all that a while ago in my book, but this part of my book is talking specifically about column space and that's what I'm confused on. How do we know that b is in the column space of A? And what does that have to do with the rank of A? It seems like you guys are saying that because the system is consistent, it follows that b is in the column space of A. But my book is first observes that b is in the column space of A, and THEN concludes that the system is consistent. – 2017-01-05
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    @dagny Ah, I think I understand what you mean now. The idea is that the echelon form of the original matrix and the augmented matrix are not very different from each other (just pretend the extra column never existed and you get the echelon form of the coefficient matrix). So to tell if their rank is the same you only need to look at the last column to see if it has an extra pivot (since the rank is the number of pivot columns). – 2017-01-05

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