Ok, I'm trying to teach myself linear algebra and having trouble understanding some the concepts...
If V is a subspace of, say R^3, then thats a collection of all vectors in some plane or some line going through the origin, correct? Then if we take the span of V, which a set of all linear combinations of the vectors in V, wouldn't that just give us the same plane or line?
2007-06-19 08:42:02 · 3 answers · asked by tzz1985 2 in Science & Mathematics ➔ Mathematics
So wouldn't V = span(V)?
2007-06-19 08:46:04 · update #1
span applies to a set, not a subspace. so, for example the span of {(1,0,0),(0,1,0)} is a plane in R^3
and for that same plane for example {(1,0,0),(0,1,0)} would be a spanning set.
but yes, if you took the set of all vectors in a subspace, the span of them would be the subspace you started off with, as by definition of a vector space it must contain any scalar multiples of any vectors within it.
and yes, your subspaces of R^3 are R^3, lines thru origin and planes through origin.
2007-06-19 08:50:01 · answer #1 · answered by Anonymous · 1⤊ 0⤋
Yes. For a *subspace*, the span is the same as the original subspace. However, if you have a *subset* that is not a subspace, the span is the smallest subspace containing that set. For example, if you take a set S to consist of two vectors in R^3, the span of S will be the plane through the origin that contains those two vectors if they are not co-linear and the line through the origin that contains those two vectors if they are co-linear.
2007-06-19 17:25:27 · answer #2 · answered by mathematician 7 · 1⤊ 0⤋
A one dimensional subspace of R^3 is spanned by a single, nonzero vector eminating from the origin. All multiples of this vector make up that subspace. A two dimensinal subspace is spanned by two linearly independent vectors (i.e. one is not a multiple of the other) eminating from the origin. All linear combinations of these vectors make up the subspace (in this case a plane through the origin). R^3 itself is spanned by three linearly independent (i.e. none of the three is a linear combination of the others) vectors. It cosists of all linear combinations of these three vectors. The only other subspace of R^3 consists of the 0 vector only (the null space). Spaces and subspaces are spanned by vectors. If the spanninjg vectors are linearly independent, they form a basis of the space.
I hope this helps.
2007-06-19 15:56:54 · answer #3 · answered by Anonymous · 0⤊ 1⤋