Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose)
Problem 136
Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.
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Add to solve later Find a Basis and the Dimension of the Subspace of the 4-Dimensional Vector Space
Problem 131
Let $V$ be the following subspace of the $4$-dimensional vector space $\R^4$.
\[V:=\left\{ \quad\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix} \in \R^4
\quad \middle| \quad
x_1-x_2+x_3-x_4=0 \quad\right\}.\]
Find a basis of the subspace $V$ and its dimension.
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