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.
What's the intuition of the transpose of a matrix?
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linear-algebra
intuition
symmetry
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0See: http://math.stackexchange.com/questions/37398/what-is-the-geometric-interpretation-of-the-transpose – 2013-09-30
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0See also: http://math.stackexchange.com/questions/484844/intuition-behind-definition-of-transpose-map – 2013-09-30