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
Linear Algebra and augmented matrix
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linear-algebra
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1This question should be generalized. – 2012-11-03
3 Answers
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- 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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0In this case, because the system has rank 5, we essentially have 5 "fundamental" (linearly inde$p$endent) equations, and 17 that are combinations o$f$ and scalar multiples o$f$ those 5. We can determine one unknown $p$er linearly independent equation. So we can determine the value o$f$ 5 unknowns, and the rest will just have to be left as parameters. – 2012-11-02
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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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The answer is 11.
Irrelevant Aside:
(Why the hell must the body be at least 30 characters long?)
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0The minimum body length is there to encourage _explanation_, rather than just a flat answer - which usually isn't helpful to the asker. – 2014-02-08