Difference between dimension and rank

In summary: But yes, if we are talking about the dimension of a vector space, then it is a different concept than the rank of a matrix. In summary, the difference between rank and dimension is that rank is an attribute of a matrix, while dimension is an attribute of a vector space. The rank of a matrix is equal to the dimension of its column space, unless the vectors formed by the columns are not independent. The dimension of a vector space is the number of linearly independent vectors that span that space. Therefore, rank and dimension cannot be compared as they are two different concepts.
  • #1
jamesweston0
26
0
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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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
Good point. I was assuming the OP was using "dimension" loosely and referring to the number of rows and columns.
 

1. What is the difference between dimension and rank?

Dimension refers to the number of independent variables or axes required to describe a space or object, while rank refers to the number of linearly independent rows or columns in a matrix. In simpler terms, dimension is related to the physical size of an object or space, while rank is related to the number of unique elements in a dataset.

2. Can dimension and rank be the same value?

No, dimension and rank are not always the same value. For example, a 3-dimensional object may have a rank of 2 if one of its dimensions is linearly dependent on the other two. Similarly, a dataset with 10 variables may have a rank of 5 if 5 of the variables are linearly dependent on each other.

3. How are dimension and rank related in linear algebra?

In linear algebra, dimension and rank are closely related. The dimension of a vector space is equal to the rank of its basis, which is the minimum number of linearly independent vectors required to span the space. In other words, the dimension of a space is determined by the number of linearly independent vectors it contains.

4. Why is it important to understand the difference between dimension and rank?

Understanding the difference between dimension and rank is important because it allows us to accurately describe and analyze the properties of objects and datasets. It also helps in solving problems related to linear transformations and matrix operations, as the dimension and rank of a matrix can affect its behavior and outcomes.

5. How can we calculate the dimension and rank of a dataset?

To calculate the dimension of a dataset, we can count the number of variables or axes required to describe it. To calculate the rank of a dataset, we can use row operations to reduce the matrix to its row-echelon form and count the number of non-zero rows. Alternatively, we can also use software such as MATLAB or Python to calculate the rank of a dataset.

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