Question

What is a $T$ in linear algebra?

Answer 1

In linear algebra, $T$ is a linear transformation.

A linear transformation is a mapping of a function between two modules that preserves the operations of addition and scalar multiplication.

Properties:

A transformation $T:V\to W$ is a linear transformation, then

• $T\left(v_1+v_2\right)=T\left(v_1\right)+T\left(v_2\right)$ , $v_1,v_2\in V$
• $T\left(k{v}_1\right)=kT\left(v_1\right)$ , where $k$ is a scalar.

Transpose of matrix:

The transpose of a matrix is also denoted by the letter $T$ of the superscript of that matrix.

Transpose of a matrix $A$ is denoted by $A^T$.

The transpose of a matrix by interchange row and column.

For exmple:

$A=\left[\begin{array}{ccc}1& -3& 4\\ 5& -2& 1\\ 6& 9& 8\end{array}\right]$

The transpose of matrix is $A^T=\left[\begin{array}{ccc}1& 5& 6\\ -3& -2& 9\\ 4& 1& 8\end{array}\right]$.