Well, I don't think that this is quite true, maybe you wanted to write something like

Just what it says: that it is a basis and, in addition, that these vectors are normalized and mutually orthogonal. Of course, a set of normalized, mutually orthogonal vectors is always linearly independent, but it need not be a basis (i.e. it may not span the entire vector space).also, what does it mean by an orthonormal basis being a basis consisting of mutually orthogonal unit vectors?

That does not follow at all. How could it? For example,does it mean that if A is a sq matrix, then the column vectors in A are orthogonal to each other?

is clearly a square matrix, but its column vectors are certainly not orthogonal to each other.

Note that from it follows that . This is because means that is the inverse of , and inverses commute...