1. Aug 20, 2011

### 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 ?

2. Aug 20, 2011

### micromass

Staff Emeritus
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.

3. Aug 20, 2011

### JamesGoh

yes thats what i meant by "true only in theory"

because the basis must be linearly independent

4. Aug 20, 2011

### micromass

Staff Emeritus
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.