# 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$$A is a symmetric matrix which elements are the inner product of each column of A. But I didn't understand the intuition of transpose. Suppose A_{m \times n}, and A transform a vector from \Bbb R^n to \Bbb R^m. But A^T transform a vector from \Bbb R^m to \Bbb R^n. What's the relationship between them? Could anyone please explain the relationship between A^T,A,the inner product and symmetric matrix. I think there would be a intuition explaination. - – Casteels Sep 30 '13 at 14:53 – Christian Blatter Sep 30 '13 at 15:19 ## 2 Answers Well, A^T is the adjoint matrix of A with respect to the ordinary inner products, i.e. A^T is the only linear mapping B such that$$\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 '13 at 11:58

One aspect of this to consider is that the transpose lets you do the same thing in different ways. The regular matrix gets multiplied by a column on the right to give your answer as a column. The transpose gets multiplied by a row on the left to give a row as the answer.

Is one better than the other? Not really, they're equivalent.

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