It is the set of vectors the are outputs of .

This is literally a linear combination of the the columns of . Where are the coefficients.

  • So the columns space is literally the span of the columns of .
  • By definition, columns space is a subset of .

Theorem 28.4

  • is a subspace of . not .
  • is the set of linear combinations of the columns of the matrix.
  • Each column of is entries tall cuz ( has rows).