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If the coefficient matrix A in a homogeneous system of 22 equations in 16 unknowns is known to have rank 5, how many parameters are there in the general solution

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Do consider sticking a homework tag on this. – Simon Hayward Nov 2 '12 at 16:06
This question should be generalized. – FrenzY DT. Nov 3 '12 at 6:04

2 Answers

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  1. So we have 5 linearly independent equations, since the system has rank 5, hence we can determine only 5 unknowns. The fact that there are 22 equations is more or less irrelevant, since 17 of them must just be linear combinations of 5 that are linearly independent.

Incidentally, is this homework?

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Somewhat. It's practice work. I don't understand how are the parameters linked to this. – Sam Nov 2 '12 at 15:57
Well, whenever there's a gap in what you can determine, you have to use parameters $k_1,k_2....$ etc. – Simon Hayward Nov 2 '12 at 16:00
In this case, because the system has rank 5, we essentially have 5 "fundamental" (linearly independent) equations, and 17 that are combinations of and scalar multiples of those 5. We can determine one unknown per linearly independent equation. So we can determine the value of 5 unknowns, and the rest will just have to be left as parameters. – Simon Hayward Nov 2 '12 at 16:02
up vote 2 down vote

I don't know whether you have seen the following concepts, but here goes:

The number of parameters is the dimension of the nullspace of the matrix.

The dimension of the nullspace of any matrix, plus the rank of the matrix, gives you the number of columns of the matrix.

And you have been given both the rank and the number of columns (since there is one column for each unknown), so you should be set!

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