`b` is not spanned by the columns of `A` if the system of three linear equations

`A(x,y,z)^T = b^T`

has no solution (ie, is inconsistent and cannot be reduced to row echelon form). That is, when gaussian elimination is completed, at least one of the rows in the transformed augmented matrix leads to an inconsistency: 1 = 0

Let `A = ` 1 1 1

2 1 1

1 1 1

(choose an `A` where at least two rows are all multiples of each other)

Then setting `b = (0, 1, 2)` (choose a `b` whose rows are such that the row ratios are not the same as the row ratios in `A`) leads to inconsistencies.

Apply gaussian elimination to the augmented matrix of the system,

`A | b`, thus:

1 1 1 | 0

2 1 1 | 1

1 1 1 | 2

Take 2*row 1 from row 2 and row 1 from row 3:

1 1 1 | 0

0 -1 -1 | 1

0 0 0 | 2

Multiply row 2 by -1 and row 3 by 1/2:

1 1 1 | 0

0 1 1 | -1

0 0 0 | 1

Gaussian elimination is complete, but this is not in row echelon form (there is no non-zero leading diagonal in the 3rd row and the row does not entirely consist of zeros) and leads to the inconsistency 1 = 0.

**So we could have**

**A = ****1 1 1 and b = (0,1,2)**

** 2 1 1**

** 1 1 1 **

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