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### What's the meaning of the transpose? [duplicate]

I don't understand the motivation of the transpose (or better yet, I haven't even seen one). It feels like just something pulled out of a hat. Thinking about it makes it seem like a product of being ...
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I know the transpose is to swap the columns and rows of a matrix. And $A^T$$A is a symmetric matrix which elements are the inner product of each column of A. But I didn't understand the intuition ... 0answers 58 views ### Geometrical interpretation of transpose of a matrix [duplicate] How can we interpret the transpose of a matrix, that represents the cordinates of a vector with respect to a basis, geometrically? 5answers 66k views ### Determinant of transpose?$$\det(A^T) = \det(A)$$Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks! 1answer 3k views ### Truly intuitive geometric interpretation for the transpose of a square matrix I'm looking for an easily understandable interpretation for a transpose of a square matrix A. An intuitive visual demonstration, how$A^{T}$relates to A. I want to be able to instantly visualize in ... 2answers 275 views ### Why is column space on the vertical in a matrix? Why is the "column space" on the vertical in a matrix? In my mind the column space is that space that the vectors in the matrix have created. I mean, for example take the equations: ... 1answer 861 views ### Geometric interpretation of matrices I'm interested in knowing some geometric interpretation of matrices. Can you suggest any lecture note or textbook or anything else about it? I've just finished an undergraduate course in linear ... 1answer 360 views ### graphical interpretation of transpose? I am working on a problem and I think that knowing if there is some sort of pattern to a transpose graphically to its normal matrix could potentially be useful. Is there any reflection, rotation, etc ... 1answer 182 views ### Geometric interpretation of the adjoint - what does it mean that the graphs are orthogonal I am trying to understand the adjoint of a linear operator geometrically. Since the graph of the adjoint can be constructed as the orthogonal complement of a "rotated" copy of the graph of the ... 0answers 68 views ### Adjoints : the mysterious linear operator! The definition of Adjoints says "Let$T$be a linear operator on an inner product space$V$. Then we say that$T$has an adjoint on$V$if there exists a linear operator$T^*$on$V$such that$\langle ...
I had to prove that det$A$ = det$A^T$. What is the geometry behind that?