Is the column space of a matrix always a full span in practice?

[JamesGoh]

In my lecture notes, the lecturer describes the column space of matrix A
as the vector space spanned by the columns of A (which means that it is assumed that the columns of A are basis )

Is this true only in theory ?

 

[micromass]

In my lecture notes, the lecturer describes the column space of matrix A
as the vector space spanned by the columns of A (which means that it is assumed that the columns of A are basis )

Is this true only in theory ?

What do you mean "only true in theory"?? The column space is by definition spanned by the columns, so it's always true.
It doesn't mean that the columns are a basis though. They might not be linear independent. For example, the column space of

$$\left(\begin{array}{cc} 1 & 2\\ 1 & 2 \end{array}\right)$$

is the span of (1,1) and (2,2). But this is not a basis of a columnspace since (1,1) and (2,2) are not linear independent.

 

[JamesGoh]

yes that's what i meant by "true only in theory"

because the basis must be linearly independent

 

[micromass]

yes that's what i meant by "true only in theory"

because the basis must be linearly independent

Well, just because they say that something spans the space, doesn't mean that this something is a basis. We can span the space without being a basis. And in general, the columns are not a basis. Only with invertible matrices do the columns form a basis.