Explain the transposing of numbers in the context of matrices.

Transposing a matrix is a fundamental operation in linear algebra that involves flipping a matrix over its diagonal. This means that the rows of the matrix become the columns and vice versa. The transpose of a matrix is denoted by an apostrophe (') or the notation $^T$ after the matrix. Let's go through the steps to transpose a matrix and discuss the properties of transposed matrices.

Step-by-Step Process of Transposing a Matrix

1. Identify the Matrix: Begin with a matrix $A$ that is of size $m \times n$, where $m$ is the number of rows and $n$ is the number of columns.

2. Create a New Matrix: Create a new matrix $A^T$ that will be the transpose of $A$. This new matrix will have the size $n \times m$, which means it will have $n$ rows and $m$ columns.

3. Switch Rows and Columns: For each element $a_{ij}$ in the original matrix $A$ (where $i$ is the row index and $j$ is the column index), place it in the position $a_{ji}$ in the transposed matrix $A^T$. In other words, the element in the $i$-th row and $j$-th column of $A$ becomes the element in the $j$-th row and $i$-th column of $A^T$.

4. Complete the Transpose: Continue the process for all elements of the original matrix until all elements have been repositioned in the transposed matrix.

Example

Let's transpose a $2 \times 3$ matrix $A$:

$$
A = \begin{bmatrix}
a & b & c \\
d & e & f \\
\end{bmatrix}
$$

Following the steps above, we create a new matrix $A^T$ that will be a $3 \times 2$ matrix:

$$
A^T = \begin{bmatrix}
a & d \\
b & e \\
c & f \\
\end{bmatrix}
$$

Notice how the first row of $A$ becomes the first column of $A^T$, the second row of $A$ becomes the second column of $A^T$, and so on.

Properties of Transposed Matrices

- Symmetric Matrix: If a matrix is equal to its transpose ($A = A^T$), then the matrix is symmetric.
- Transpose of a Transpose: The transpose of the transpose of a matrix is the original matrix ($(A^T)^T = A$).
- Transpose of a Sum: The transpose of a sum of two matrices is equal to the sum of their transposes ($(A + B)^T = A^T + B^T$).
- Transpose of a Product: The transpose of a product of two matrices is equal to the product of their transposes in reverse order ($(AB)^T = B^T A^T$).

Transposing a matrix is a simple yet powerful tool in linear algebra that is used in various applications, including solving systems of linear equations, performing matrix decompositions, and working with tensors in higher dimensions.