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Question about column spaces

  1. Aug 20, 2011 #1
    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. jcsd
  3. Aug 20, 2011 #2
    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

    [tex]\left(\begin{array}{cc} 1 & 2\\ 1 & 2 \end{array}\right)[/tex]

    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.
  4. Aug 20, 2011 #3
    yes thats what i meant by "true only in theory"

    because the basis must be linearly independent
  5. Aug 20, 2011 #4
    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.
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