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My textbooks say that when a solution, x, is found to Ax=b it has a particular solution, x_0, such that A*x_0=b which is then combined with other solutions from the null space, n_i, such that A*n_i=0.

However, when playing about with this I seem to have come across a problem.

for the system:

|1 3 3 2| |u| |1|

|2 6 9 5| |v| = |5|

|-1 -3 3 0| |w| |5|

|y|

I get an LU factorisation of A to be:

| 1 0 0 | | 1 3 3 2 |

| 2 1 0 | | 0 0 3 1 |

|-1 2 1 | | 0 0 0 0 |

when solved for x this gives x_0 as:

|-2 |

| 0 |

| 1 |

| 0 |

and n_1 and n_2 as:

|-3 | | -1 |

| 1 | | 0 |

| 0 | |-1/3|

| 0 | | 1 |

Using the rows of U as a basis for the row space of A, the particular solution, x_0, cannot be formed, so does not lie in the row space of A as it should.

Have I done something wrong or is my understanding incorrect (is the "row space component of x" my books talk about not the same as x_0...)?

Many hanks in advance ;)

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# Solutions to Ax=b lie in row space?

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