# what's the intuition of the transpose of a matrix

I know the transpose is to swap the columns and rows of a matrix. And $A^T$$A is a symmetric matrix which elements are the inner product of each column of A. But I didn't understand the intuition of transpose. Suppose A(m x n), and A transform a vector from \Bbb R^n to \Bbb R^m. But A^T transform a vector from \Bbb R^m to \Bbb R^n. What's the relationship between them? Could anyone please explain the relationship between A^T,A,the inner product and symmetric matrix. I think there would be a intuition explaination. - ## 1 Answer Well, A^T is the adjoint matrix of A with respect to the ordinary inner products, i.e. A^T is the only linear mapping B such that$$\langle Av,w\rangle = \langle v,Bw\rangle$$for all$v\in\Bbb R^n$and$w\in\Bbb R^m$. You can easily see it if you verify it on the standard bases, noting that$\langle u,e_i\rangle$gives the$i$th coordinate of$u\$.

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 thank you. But how to use this conclusion? Is there any geometry explaination? – maple Jan 10 at 11:58