From the method given to me in my textbook, to find the spanning set of a null space. we row reduce our matrix to find the basic solutions of the parameters. However, it seems like the process is exactly the same when trying to find our basis for the null space. Since the definition of a basis is that it spans the set and it is linearly independent, is the basis of a null space the same as its spanning set?
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$\begingroup$ There's not a single spanning set! The null space is a spanning set, for instance $\endgroup$– BernardNov 28, 2019 at 21:34
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$\begingroup$ imgur.com/a/YHOXzar This was an example given to us to find the spanning set. But wouldnt the basis also be {(1,1,1,0),(-1,0,0,1)} for this particular null space? $\endgroup$– Mio ChibanaNov 28, 2019 at 21:35
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$\begingroup$ To find a spanning set… $\endgroup$– BernardNov 28, 2019 at 21:37
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$\begingroup$ A basis is a linearly-independent spanning set. $\endgroup$– amdNov 28, 2019 at 21:40
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$\begingroup$ There are also spanning sets that are not linearly independent, and therefore are not bases. But the usual row reduction algorithm always produces a basis. $\endgroup$– Robert IsraelNov 28, 2019 at 21:43
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1 Answer
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Every basis is a spanning set, but not every spanning set is a basis. The difference is that a spanning set need not be linearly independent.
More exactly, a basis is a minimal spanning set, that is, if you remove any vector from it, the resulting set no longer is a spanning set.
