Non-invertible operators

Question

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

Certainly - both questions could be answered in a purely mathematical context. — Shivam Sarodia, Mar 19 '14 at 16:59
You should really split this question up into two. The first question is not quite clear: are you asking "are there linear operators on some vector space whose matrix representation is singular?"? If so, you should make it that precise. — E.P., Mar 19 '14 at 17:00

score 2 · Accepted Answer · answered Mar 19 '14 at 17:01

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 $a\neq b\in\mathbb{R}$.

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