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.

 

[blue_raver22]

Yes, this statement is true in theory. In practical applications, the column space of a matrix A is often referred to as the range or image space of A. It represents all possible linear combinations of the columns of A, and can be visualized as a subspace of the vector space in which A operates. In theory, the columns of A are assumed to be linearly independent and form a basis for the column space. However, in reality, the columns of A may not always be linearly independent, in which case they would not form a basis for the column space. In such cases, the column space may not be the full span of the columns of A, but rather a smaller subspace. Therefore, while the concept of column space and its definition remain the same in theory, the actual dimensions and properties of the column space may vary in practical applications.