What is the geometrical interpretation to $AX = 0$ when $det(A) = 0$? I read the first chapter of hoffman and from there I know that algebraically how to find the infinite solutions to $AX=0$ when $det(A)=0$.
However I am interested in seeing the geometric intuition behind it.
Assume that $A$ is a $2 \times 2$ matrix then  $$A=\begin{bmatrix}a & b \\c&d \end{bmatrix}$$
If the $\det(A) = 0$ then assume that the second row is a multiple of the first row so $(c,d) = k(a,b)$.
This doesn't help me to see as to there will exist a solution at the first place(geometrically) or if this doesn't happen then why is the solution unique(geometrically)?
Edit 1:
When I am doing $AX = 0$ I am trying to find the solution to my question :
When does the two lines $ax_1 + bx_2 = 0 $ and $cx_1 + dx_2 = 0$ meet. They always meet at the point $(0,0)$. Now the two lines can be coincident(if they are linearly dependent) or they can only meet at $(0,0)$ (in which case they are linearly independent). Now if they are coincident then the two lines lie on top of each other and so they have infintely many points satisying their equations.
Are there any other possible cases?
 A: What you describe in the edited question is one perfectly good geometric interpretation of the matrix equation. It can serve you well -- but will serve you better if you avoid thinking of it as "the" geometric interpretation that you need to hold on to in all cases.
In particular, as you work your way through linear algebra, you will learn that a particularly strong idea is to view the matrix not as two independent linear equations, but as a representation of a linear transformation between two abstract two-dimensional vector spaces. You have a wide choice of expressing such as transformation using different coordinate systems when writing down as a matrix -- and if you switch to a different coordinate system for the output of the linear transformation the two apparently-separate equations into a mixture and the lines you're imagining now cease to have individual significance. Conversely, when you have a matrix to start with, it can often make it easier to "see the forest for trees" to forget about its individual elements for some time and just think of it as some linear transformation -- in many cases you'll be able to understand better what the matrix does by switching coordinates to one where the matrix is simpler.
(The lines in your current picture are simply the preimages of the output coordinate axes, when mapped backwards through the linear transformation. If you use a different output coordinate system, its axes will come from different locations in the input space. (However, beware that this connection doesn't generalize directly to higher dimensions, so that way of thinking is not any "ultimate truth" either)).
