Difference between dimension and rank

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Discussion Overview

The discussion revolves around the concepts of rank and dimension in the context of linear algebra, specifically regarding matrices and vector spaces. Participants explore the definitions and implications of these terms, seeking clarity on their differences and relationships.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants express confusion about the difference between rank and dimension, noting that a book states the rank of a matrix is the dimension of the column space.
  • One participant argues that rank and dimension cannot be compared directly, as rank is an attribute of a matrix while dimension pertains to vector spaces.
  • Another participant clarifies that every vector space has a dimension, and the dimension of the column space of a matrix is referred to as the rank of that matrix.
  • There is a suggestion that the original poster may be conflating "dimension of the column space" with "number of columns," highlighting that rank can be less than the number of columns if the columns are not independent.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the comparison between rank and dimension, with some asserting they are equivalent in specific contexts while others maintain they are fundamentally different concepts.

Contextual Notes

There is a potential misunderstanding regarding the definitions of rank and dimension, particularly in relation to the independence of columns in a matrix. The discussion reflects varying interpretations of these terms.

jamesweston0
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Hey all.

I know this is a basic concept but I don't really understand it. I don't get what the difference between rank and dimension is. According to my book, the rank of a matrix is the dimension of the column space. Does that not imply that they are the same, unless the question specifically states they are different? And how would I be able to even tell if they are different unless it tells me?

Confusing to me to say the least.

Thanks.
 
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jamesweston0 said:
Hey all.

I know this is a basic concept but I don't really understand it. I don't get what the difference between rank and dimension is. According to my book, the rank of a matrix is the dimension of the column space. Does that not imply that they are the same, unless the question specifically states they are different? And how would I be able to even tell if they are different unless it tells me?
Tell how what are different? Rank and dimension of the column space? They are never different! That's precisely what your book says: "the rank of a matrix is the dimension of the column space".

Confusing to me to say the least.

Thanks.
Why confusing? Your book says something and you are asking "how do I know this is true?" Why should you doubt it?

Perhaps you are confusing "dimension of the column space" with "number of columns". The dimension of the column space is equal to the number of columns if and only if the vectors formed by the columns are independent. If not, then the rank will be less than the number of columns.
 
The rank is an attribute of a matrix, while dimension is an attribute of a vector space. So rank and dimension cannot even be compared.
 
Every vector space has a dimension. The dimension of a particular vector space, namely the column space of a matrix, is what we call the rank of that matrix.
 
Good point. I was assuming the OP was using "dimension" loosely and referring to the number of rows and columns.
 

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