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

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Discussion Overview

The discussion centers on the concept of matrix rank, specifically what it means for a matrix to have rank 0. Participants explore the implications of rank 0 in terms of linear maps and vector spaces, as well as the relationship between non-zero elements and matrix rank.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant states that the rank of a matrix is the dimension of the image of the linear map it represents, concluding that a rank 0 matrix has an image that is the zero vector space.
  • Another participant suggests that a matrix with at least one non-zero element must have a rank of at least 1, prompting further inquiry into this assertion.
  • A different participant elaborates that a zero-dimensional row space consists solely of the zero vector, indicating that if any row contains a non-zero entry, the row space would have a dimension of at least one.

Areas of Agreement / Disagreement

Participants express differing views on the implications of non-zero elements in relation to matrix rank, indicating that there is no consensus on the broader implications of these definitions.

Contextual Notes

The discussion includes assumptions about the definitions of vector spaces and matrix rank that may not be universally agreed upon, and there are unresolved implications regarding the transition from rank 0 to rank 1.

sauravrt
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What does it mean for a matrix to have rank 0 ( zero) ?
 
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The rank of a matrix is the dimension of the image of the linear map it represent. Since the only vector space of dimension 0 is the vector space denoted 0 consisting of only one elements (namely, 0), to say that a matrix is of rank 0 is to say that the image of the linear map it represents is the vector space 0.

You can convince yourself that, in turn, this implies that the matrix is the matrix 0 (the matrix having 0 in all its entries...)
 
Thanks quasar987.

So a matrix with atleast one non-zero element will have atleast rank 1 ?
 
What do you think? And why?
 
Yes, the rank of a matrix is the dimension of the row space.

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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