What does it mean for a matrix to be nearly singular?

I am currently enrolled in numerical analysis course, and a term I have not heard of came up: nearly singular matrix.

I know that a non-singular matrix is one where the column vectors are linearly independent.

My guess is that a nearly singular matrix is one where one of the column vectors $v$ can be "almost" represented as a linear combination of other column vectors of the matrix, but I am really not sure.

Also, why are we concerned with nearly singular matrix in numerical computation? (The term never came up in my linear algebra class, so I assume it has no significance in terms of theoretical aspect.)

1 Answer

A more common term for nearly singular matrix is ill-conditioned. If a matrix has a large condition number then it is called ill-conditioned. Computations involving ill-conditioned matrices are usually very sensitive to numerical errors.