What Is the Relationship Between Matrix Rank and Transpose?

TL;DR

The rank of a matrix A is equal to the rank of its transpose, which means they have the same number of pivot entries in reduced row echelon form. This relationship holds because the dimension of the column space of A is equivalent to the dimension of the row space of the transpose, and both are determined by the number of linearly independent vectors in their respective spaces.

Transcript

A couple of videos ago, I made the statement that the rank of a matrix A is equal to the rank of its transpose. And I made a bit of a hand wavy argument. It was at the end of the video, and I was tired. It was actually the end of the day. And I thought it'd be worthwhile to maybe flush this out a little bit. Because it's an important take away. It'... Read More

Key Insights

• 😜 The rank of a matrix A is equal to the rank of its transpose.
• 👾 The dimension of the column space of A transpose is the same as the row space of A.
• 🤨 The basis for the row space of A is formed by the pivot rows in the reduced row echelon form of A.
• 🔄 The number of basis vectors for the column space of A transpose can be determined by counting the pivot entries in the reduced row echelon form of A.
• 😜 The rank of A is equal to the number of pivot entries in the reduced row echelon form of A, which is also the rank of A transpose.
• 🤨 The column space of A transpose and the row space of A are equivalent and can be represented by the span of the rows of A.

Questions & Answers

Q: What is the rank of the transpose of a matrix?

The rank of the transpose of a matrix A is equal to the dimension of the column space of A transpose, which is the same as the row space of A.

Q: How can we determine the number of basis vectors for the column space of A transpose?

The number of basis vectors for the column space of A transpose can be determined by counting the pivot entries in the reduced row echelon form of A.

Q: What is the relationship between the column space of A transpose and the row space of A?

The column space of A transpose and the row space of A are equivalent and can be represented by the span of the rows of A.

Q: What is the basis for the row space of A?

The pivot rows in the reduced row echelon form of A form a suitable basis for the row space of A.

Summary & Key Takeaways

• The rank of the transpose of matrix A is equal to the dimension of the column space of A transpose, which is the same as the row space of A.

• The number of basis vectors needed for the column space of A transpose is the same as the number of basis vectors for the row space of A.

• The pivot rows in the reduced row echelon form of A form a suitable basis for the row space of A, which is also the column space of A transpose.

• The rank of A is equal to the number of pivot entries in the reduced row echelon form of A, which is also the rank of A transpose.