Completeness of basis in quantum mechanics

[wowowo2006]

In a QM course,
I learn that an operator can be represented by basis vectors
If the basis vector is complete, the following relation holds
There exist coefficient Mij such that
Sigma Mij |i > < j|. = I , |i> is the basis! and I is the identity matrix

But isn't that in linear algebra
We call the set of basis is complete when
Any vector can be expressed into their linear combination

I wonder why here we seems have 2 definition of completeness

 

[unscientific]

I recommend reading of Sakurai's Book on QM - read the introductory chapters.

 

[BvU]

I wonder why here we seems have 2 definition of completeness

There is only one definition if the two renderings are equivalent, right ? And they are equivalent!