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The transpose of a matrix is a linear algebra operator that flips a matrix diagonally, that is, it reverses the row and column indices of matrix A by producing another matrix, generally represented by${\mathit{A}}^{\mathbf{T}}$among other notations).

For example, the transpose of $\left[\begin{array}{cc}0& 1\\ 4& 7\end{array}\right]is\left[\begin{array}{cc}0& 1\\ 4& 7\end{array}\right]$

The transpose that operates on matrices needs to be determined.

The transpose of a matrix is a linear operator.

A linear operator O satisfies the condition.

$$\begin{gathered} O(A+B)=O\left(A\right)+O\left(B\right) \\ O\left(kA\right)=kO\left(A\right) \end{gathered}$$

The transpose of a matrix is a mapping.

$A↦A^T$

It can be written in terms of matrix elements $a_{ij}$.

role="math" localid="1658994061609" $a_{ij}↦a_{ji}$

If the sum of two matrices and is given, then the element of $A+B=C$ will be given as role="math" localid="1658994227909" $a_{ij}+b_{ij}+c_{ij}$ . The transpose takes the element $c_{ij}to{c}_{ij}$, which can be written as $a_{ij}+b_{ij}$, which are the elements of $A^{T}and,B^T.$ .

In matrix terms, addition is done as $C^T=A^T+B^T.$.

The first condition is satisfied. The matrix has elements , and the transpose simply maps them to $k{a}_{ji}$, which in terms of matrices gives$(kA{)}^T="k{A}^T$ ,

which means that the transpose satisfies both conditions of a linear operator.

Therefore, the transpose of a matrix is a linear operator