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What does it mean for a matrix to have rank 0 ( zero) ?

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- #1

- 15

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What does it mean for a matrix to have rank 0 ( zero) ?

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- #2

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You can convince yourself that, in turn, this implies that the matrix is the matrix 0 (the matrix having 0 in all its entries...)

- #3

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Thanks quasar987.

So a matrix with atleast one non-zero element will have atleast rank 1 ?

So a matrix with atleast one non-zero element will have atleast rank 1 ?

- #4

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What do you think? And why?

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If a matrix has a zero-dimensional row space, it consists of a single vector - the zero vector. The space consisting of the zero vector only has dimension zero.

If a vector had an entry besides 0, then that row would not be the zero vector. Then the row space would include inifinitely many vectors corresponding to all scalar multiples of that vector... and have dimension at least one.

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