Difference between dimension and rank

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Rank and dimension are closely related concepts in linear algebra, specifically regarding matrices and vector spaces. The rank of a matrix is defined as the dimension of its column space, meaning they are effectively the same in this context. However, confusion arises when considering the number of columns, as the rank can be less than the number of columns if the vectors are not independent. It's important to distinguish that rank is a property of a matrix, while dimension pertains to vector spaces. Thus, while rank and dimension are linked, they serve different roles in understanding linear algebra.
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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