The transpose of a matrix is the result when you represent the columns as rows. The transpose of a matrix A can be represented as $ A^T $. For example,

$ \begin{bmatrix} 1 & 2 & 3\\ 0 & 5 & 10\\ 0 & 8 & 7\end{bmatrix}^T=\begin{bmatrix} 1 & 0 & 0\\ 2 & 5 & 8\\ 3 & 10 & 7\end{bmatrix} $

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Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

Dr. Paul Garrett