Suppose that we are given a LP in canonical form, that is in the form $\{x \in \mathbb{R}^d |\ Ax \geq b \}$ and that we want to convert it to an equivalent LP in standard form $\{x \in \mathbb{R}^k \ |\ A'x = b, x \geq 0 \}$, $k\geq d$.
The new standard form polytope will usually lie in a higher-dimensional space due to the introduction of the slack variables.
I am wondering about how this transformation will affect the geometry of the original polytope.
So my question is, will it have a different shape which will be isomorphic to the original one or will it maintain its shape and it will just be embedded in a higher-dimensional space?
Thank you for your time.
Different shape in the higher-dimensional space, and if you project down from the higher dimensional space you'll get back your original polyhedron (shape-wise, might have different representation as a matrix). You can satisfy yourself of this by using Fourier-Motzkin elimination.