For simplicity let $T:R^{n}\rightarrow R^{n},$ be a linear operator and $[A]_T$ be the matrix of operator $T.$ Then matrix of adjoint $T^{\times}$ operator of $T$ is given as $$[A]_{T^{\times}}=[A]_{T}^t$$ where $t$ denotes transpose of a matrix.
My Question is that then from above equality the notions of transpose of a matrix and adjoint of a matrix are same, they why we use separate names?? while in some textbooks adjoint of a matrix is referred to the transpose of coefficient matrix.
On
First, the adjoint is defined as a conjugate transpose for complex operators, but this reduces to a pure transpose in the real case.
Also, the transpose is only defined for finite matrices (finite-dimensional operators). The adjoint operator can be defined for infinite-dimensional operators and even more general things, like Hilbert spaces. Hence it makes sense to call it by a different name.
In the context of complex vector spaces, they are different: the adjoint matrix is the conjugate of the transpose matrix.