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Whenever we see a matrix $A=\bigl( \begin{smallmatrix} 3 & 2 \\ 1 & 2 \end{smallmatrix} \bigr)$ and $v=(3, 2),$ we can visualize that $(3, 2)$ represent the coordinates of $\mathbf i$ vector and $(1,2)$ represents the $\mathbf j$ vector. And to visualize $A$ as a whole, we can see it as a matrix that transforms can the vector $v.$ when we multiply it.

So, we can visualize $A^{-1}$ as a matrix that undoes the transformation by $A.$ Similarly, the determinant of $A$ gives the area.

How can I visualize the transpose of a matrix as well as matrices that have a complex number as an element? Thank you.

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    Maybe not the most complete explanation but for a vector space with an inner product, we have $\langle x, Ax'\rangle = \langle A^Tx, x'\rangle$. So $A^T$ transforms the other vector $x$ in a way that it has the same inner product with $x'$ as the transformed $x'$ has with $x$. – user1936752 Jul 21 '20 at 14:58
  • Since left or right multiplication of matrices is arbitrarily chosen I interpret the transpose as exactly the same as the original matrix just written sideways. This would be analogous to using row vectors instead of the canonical column vectors - they represent the same object we just write them down differently. – CyclotomicField Jul 21 '20 at 15:32
  • Sorry I didn't got your point.In general my question is ,see when I multiply something with A-1 it undoes the transformation.So A-1 has a significance. While in Maths books A' is everywhere means it too must have some significance ,that could be visualized in column or row space maybe. Please explain me what is that significance.Thank you – Newton Nadar Jul 22 '20 at 04:23
  • Ya very much .But looks I lack the basics init. Please suggest a book for linear transformation which help for geometric intiuation .After that I will delete this question .Its just a copy.Thank you. – Newton Nadar Jul 22 '20 at 13:40

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