Question about row space basis and Column space basis

[PsychonautQQ]

Say a subspace S of R^3 is spanned by a basis = <(-1,2,5),(3,0,3),(5,1,8)>

By putting these vectors into a matrix and reducing it to rref, a basis for the row space can be found as <(1,-2,-5),(0,1,3)>. Furthermore, the book goes on to say that this basis spans the subspace S.

Cool, not suprising.

My question then is if the basis for the column space also spans S. If so, that means span(basis of column space) = span(basis of row space)?

Why doesn't my book say this straight up!?

 

[Stephen Tashi]

Why doesn't my book say this straight up!?

It isn't clear what you mean by "this". Are you asking a question about the one particular problem or are you asking about a general statement that applies to all matrices? If it's a general statement, can you state the conjecture in mathematical form? (If ... such-as-such then ...so-and so.)

Not all matrices are square. The span of a set of row vectors might not be in the same vector space as the span of a set of column vectors.

 

[WWGD]

For the relationship between the two bases, see, e.g.:

http://en.wikipedia.org/wiki/Fundamental_theorem_of_linear_algebra