Yes, let's prove that the rank of a matrix $A$ is the same as the rank of its transpose $A^T$.

Definitions:

• The rank of a matrix is the dimension of its column space (or equivalently, its row space).
• The transpose of a matrix $A$, denoted $A^T$, is a new matrix where the rows and columns of $A$ are swapped.

Proof:

Let $A$ be an $m \times n$ matrix, and let $A^T$ be its transpose, which will be an $n \times m$ matrix.

1. Column space of $A$ and row space of $A^T$:
   • The column space of $A$ is the span of the columns of $A$. It is a subspace of $\mathbb{R}^m$.
   • The row space of $A$ is the span of the rows of $A$. It is a subspace of $\mathbb{R}^n$.
   • The column space of $A^T$ is the span of the rows of $A$, which is the row space of $A$.
   • The row space of $A^T$ is the span of the columns of $A$, which is the column space of $A$.

Thus, the column space of $A$ is the same as the row space of $A^T$, and the row space of $A$ is the same as the column space of $A^T$. This establishes that the dimensions of the column space of $A$ and the row space of $A^T$ are the same.

2. Rank of $A$ and rank of $A^T$: The rank of a matrix is the dimension of its column space (or row space, as both are equal in rank). Since the column space of $A$ and the row space of $A^T$ have the same dimension, we conclude that: $\text{rank}(A) = \text{rank}(A^T)$

Conclusion:

The rank of a matrix $A$ is equal to the rank of its transpose $A^T$.

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Would you like a deeper explanation of any specific step, or is there another concept you'd like to explore? Here are 5 related questions that could help extend your understanding:

1. How does the rank of a matrix relate to its nullity (the dimension of the null space)?
2. Can you prove the rank-nullity theorem using the result from this proof?
3. What are some methods to compute the rank of a matrix?
4. How does the rank of a matrix change under row or column operations?
5. What is the rank of a matrix product $AB$ in terms of the ranks of $A$ and $B$?

Tip: When calculating the rank of a matrix, performing Gaussian elimination (row reduction) is a helpful way to find the rank by transforming the matrix into row echelon form.