Is it the case that the adjoint of a matrix is just its transpose- the definition is based off inner products. Does it matter whether it is a complex or real inner product?
2026-03-29 03:44:37.1774755877
Adjoint linear transformations
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For a linear transformation $T:V\to W$, the matrix of its adjoint $T^*:W^*\to V^*$ will be the (conjugate) transpose of its matrix if you’re using dual bases for the vector spaces. When you’re working in $\mathbb R^n$ with the standard basis and the usual Euclidean scalar product, which is what it looks like you’re asking about, this condition is satisfied.