Currently, I am learning 3D Vision. I am trying to visualize projective space. I am not Mathematician so please pardon me if I have missed out subtle concreteness.
Q-1 In $P^2$ we can visualize each point in $P^2$ as a line passing through the origin through in $E^3$ and each line in $P^2$ as passing through the origin in $E^3$. Can we extend the same object visualization for $P^3$ i.e each point in $P^3$ can be visualized as a line in $E^4$ passing through the origin and each line in $P^3$ can be visualized as a plane in $E^4$ passing through the origin?
Q-2 How can you prove that usage of homogeneous coordinate makes the perspective transformation linear? Can you provide an informal insight of the proof?
Q-1: Yes, this generalizes to higher dimensions. Every $k$-dimensional projective subspace of $\mathbb P^d$ can be visualized as a $(k+1)$-dimensional linear (i.e. passing through the origin) subspace of $\mathbb E^{d+1}$.
However, this is not as powerful as it seems at fist glance. One of the beautiful things about $\mathbb P^2$ is that you can describe a point by a vector (the direction of the line in $\mathbb E^3$) and a plane by a vector (the normal of the plane in $\mathbb E^3$). In both cases the length of the vector does not matter, which is the key property of homogeneous coordinates. A lot of elementary operations boil down to intuitive manipulations of $3$-vectors.
A $2$-plane in $\mathbb E^4$ is described by two direction vectors, or two normal vectors, and either of them might be scaled independently. So this whole nice representation tends to break down a bit. The best approach for recovering it would be Plücker coordinates. These would e.g. represent a line in $\mathbb P^3$ as a vector with $6$ elements satisfying a certain quadratic constraint. But there are various conventions regarding notation, order of elements and signs, so be careful comparing texts. Personally I'm used to the convention from J. Richter-Gebert's Perspectives on Projective Geometry Chapter 12, but that's due to lectures by the author.
So points and hyperplanes in higher dimensions are fine, but everything in between may require more work.
Q-2: This depends as how you define a projective transformation. I tend to see it as a non-singular matrix applied to a homogeneous coordinate vector, in which case it's linear by definition.
A more geometric property would be defining a projective transformation by it's property that it preserves collinearity: if three points are on a single line before the transformation, then so are their images under the transformation. But technically that's the definition of a collineation. While over the real numbers every collineation is a projective transformation, over e.g. the complex numbers this is no longer true: taking complex conjugates of all coordinates will preserve collinearity but is not a linear transformation.
Your comment suggests that the main question is how homogeneous coordinates fit in with operations using divisions. For this purpose, let's do homogenization, projective transformation and dehomogenization one after the other:
$$\begin{pmatrix}x\\y\end{pmatrix}\rightarrow \begin{pmatrix}x\\y\\1\end{pmatrix}\mapsto \begin{pmatrix}a&b&c\\d&e&f\\g&h&k\end{pmatrix} \cdot\begin{pmatrix}x\\y\\1\end{pmatrix}= \begin{pmatrix}ax+by+c\\dx+ey+f\\gx+hy+k\end{pmatrix}\rightarrow \frac{1}{gx+hy+k}\begin{pmatrix}ax+by+c\\dx+ey+f\end{pmatrix}= \begin{pmatrix}\frac{ax+by+c}{gx+hy+k}\\[1ex] \frac{dx+ey+f}{gx+hy+k}\end{pmatrix}$$
So you can express a transformation where the resulting coordinates are fractions, as long as the individual numerators and the common denominator are linear in the input coordinates when expressed without homogeneous coordinates.