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I know that row rank of a matrix is the number of row vectors that span row space of the matrix. Column rank can be similarly defined.

I also know that row rank = column rank and have learned the proof from the book of linear algebra written by Atindra Mohun Gun.

But will this proof of row rank and column rank be sufficient to show that rank of a matrix is unique ?

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2 Answers

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Before we can talk about matrix rank, we have to talk about row rank, which is the dimension of row space of the matrix. You should have proven that row rank is unique.

Similarly, column rank is unique.

From the result that row rank is equal to column rank, we can then talk about matrix rank. Of which, the uniqueness is due to uniqueness of row rank.

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1. You know the definition of row rank of a matrix.

2. You know the definition of column rank of a matrix.

3. You prove that row rank of a matrix=column rank of a matrix.

4. Now you define rank of a matrix=column rank of a matrix.

Then by definition it is unique.

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