Non-invertible operators

Question

1. Can the matrix representation of some linear operators on some vector space be singular?

Answer 1

Can singular matrices (with determinant=0) represent linear operators in a vector space?

Yes. The definition of a linear operator includes no restrictions on invertibility.

Is the cardinality of two segments of different length of the real line is same or different?

It is the same, namely $|[a,b]|=|\mathbb{R}|=2^{\aleph_0}$ for any $a\neq b\in\mathbb{R}$.