If a matrix can be written as a linear combintation infinitely many ways.. does that mean its linear independent ??

av1+bv2+cv3=M

I know to qualify for Independence.. a=b=c=0

and dependence a,b,c must be scalars but not all should be 0.

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- March 16th 2010, 07:55 PMKhonics89Linear independent question
If a matrix can be written as a linear combintation infinitely many ways.. does that mean its linear independent ??

av1+bv2+cv3=M

I know to qualify for Independence.. a=b=c=0

and dependence a,b,c must be scalars but not all should be 0. - March 16th 2010, 08:52 PMkaylakutie
Don't quote me because I just learned this stuff like last week and don't fully understand it yet... but here's my best guess.

I think those vectors would span M then, given it is a 3x3 matrix. Meaning a particular combination of those vectors can produce any 3x3 matrix M.

To be independent, none of the vectors in a subset can be linear combinations of each other. If one of them is a multiple of another or linear combination of other ones, it is redundant and thus provides no new information - so it's dependant. In order to span the 3x3 matrix, the 3 vectors must be independent. But if it were a 2x2 matrix, I believe it can still span while being linearly dependent, provided only one of the vectors is redundant. - March 17th 2010, 09:27 PMChris11
Linear independence: c1v1+c2v2+...+cnvn has only the trivial solution IE c1=...=cn. SO no.