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In one lectures on linear algebra, my professor wrote on the first day:

Let $L$ be in a linear map $L: X \to Y$, then $Im(L) \oplus Ker(L^*) = Y$

On the second day, he wrote: $L: X \to Y$, then $Im(L) \oplus Ker(L^T) = Y$

I am a little bit confused by the mixture of adjoint and transpose. I suspect the transpose is used by assuming that $L$ is the matrix representation of the transform, whereas adjoint is just a general symbol to denote the adjoint of a linear transform and does not induce a transpose.

Can someone clarify this concept for me? THanks!

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  • $\begingroup$ That's typically how I interpret it, but I've seen others use the words "transpose of a linear transformation" and "adjoint of a matrix" (meaning transpose, not adjugate) before. It seems the words are somewhat interchangeable. $\endgroup$ – user137731 Oct 30 '15 at 15:16
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Suppose you have a linear transformation $L : X \to Y$ where $X$ and $Y$ are vector spaces endows with inner products $\langle \cdot,\cdot \rangle_X$ and $\langle \cdot,\cdot \rangle_Y$, respectively. Then the adjoint of $L$ is a linear map $L^* : Y \to X$ such that $$ \langle x, L^*(y) \rangle_X = \langle L(x), y \rangle_Y \quad \forall x \in X, \forall y \in Y. $$ Now, if $X$ and $Y$ are real vector spaces and $L$ is represented by a matrix $A$ with respect to a choice of orthonormal bases, then $L^*$ is represented by the transpose $A^T$ with respect to the same bases.

On the other hand, this may not hold over other fields. For example if $X$ and $Y$ are complex vector spaces, then $L^*$ is represented by the Hermitian transpose $A^{\dagger} = \overline{A^T}$.

In conclusion, when we talk about the adjoint of a matrix $A$, which represents a linear map $L$ with respect to some choice of orthonormal bases, we mean the matrix representing the adjoint map $L^*$ with respect to the same bases, usually denoted by $A^*$.

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