Is there a nice geometric interpretation of reversing the components of a vector? So given some [imath]\begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix}[/imath], I want to visualize how it relates to [imath]\begin{bmatrix}x_3 \\ x_2 \\ x_1\end{bmatrix}[/imath]. Is there a specific linear transformation that accomplishes this reversal? I have tried graphing a few vectors and their "algebraic inverses," and all I have noticed so far is that the length of the original vector is preserved.
Ultimately I am interested in a geometric interpretation of performing this action followed by taking a dot product with another vector (which in and of itself has a geometric interpretation of projection followed by scaling), as this is very close to a discrete convolution. Discrete convolution has its own geometric interpretation defined on functions of a discrete variable, but these are basically identical to vectors, so I was wondering what the operation would look like in geometric vector space.
Ultimately I am interested in a geometric interpretation of performing this action followed by taking a dot product with another vector (which in and of itself has a geometric interpretation of projection followed by scaling), as this is very close to a discrete convolution. Discrete convolution has its own geometric interpretation defined on functions of a discrete variable, but these are basically identical to vectors, so I was wondering what the operation would look like in geometric vector space.