span

nothGing

can explain to me what is mean of span?
from book, it say "every vector in the space can be expressed as linear combination of the vectors, then it called the span of vector."
but i still cant catch its meaning and concept.

HallsofIvy

Okay, do you know what linear combination means? The two things you can do with vectors are add them and multiply them by numbers. For example, if my set of vectors is {<1, 0, 1> , <1, 1, 0>} then all linear combinations of them are things like a<1, 0, 1>+ b<1, 1, 0>= <a, 0, a>+ <b, b, 0>= <a+b, b, a>. Things like that are in the "span" of {<1, 0, 1>, <1, 1, 0>}. As long as I stay in that span, I can just focus on <1, 0, 1> and <1, 1, 0> and all other vectors (in that span) are taken care of! I can, by the way, immediately see that <3, 2, 1> is in that span and that <3, 2, 2> is not (look at whether the first component is the sum of the other two).

There are an infinite number of vectors in any vector space and we would like to be able to write all of them in terms of some smaller set. If we can then we say that set "spans" the entire vector space.

nothGing

the smaller set in R^2 is {<1,0> , <0,1>}, this set "spans" entire R^2, correct?

how can span related to linear independence of the set of vectors, S in a vector space, V ?
if the set of vectors, S is "linearly independent or linearly dependent", then S "must" be a span of V.
or should say S spans V. correct?

under what situation or condition, S is not a span, ie the set of vectors S cant express as a linear combination of vectors in S?
because seen like all set of vectors in S can express as linear combination of vectors in S.

HallsofIvy

Saying that a set spans a vector space means that any vector in the vector space can be written as a linear combination of vectors in the set- possibly in more than one way. For example, the set {<1, 0>, <0, 1>, <1, 1>} spans R² because any vector can be written <a, b> = a<1, 0>+ b<0, 1>. In fact, <a, b>= (a-x)<1, 0>+ (b-x)<0, 1>- x<1, 1,> for any number x.

If a set of vectors is independent, it may not be that all vectors can be written as a linear combination of vectors in the set, but those that can be can be written in only one way.

It is not too hard to show that if a set of vectors spans the space then no larger set can be independent and that if a set of vectors is independent, then no smaller set can span the set. There always exist a specific number of vectors that can both be independent ans span the space.

If a set of vector both spans the space and is independent, then every vector in the vector space can be written as a linear combination of vectors in the set in exactly one way. That is very nice!

nothGing

"span" is still obscure to me.
but in overall, im more nearer to it.
your explaination really useful.
thank you. ^^

Mark44

The "span" of a set of vectors in a space is somewhat related to the meaning of this word in English. In a non-mathematics context, we can talk about a bridge that spans (or goes across) a river. In the mathematical context, you might say we can say that a certain set of vectors "goes across" a space, in the sense that I can use the vectors in the set to get to a specific vector. How I use the vectors is to form a linear combination of them, which is the sum of scalar multiples of the vectors.

For example, consider this set of vectors in R³: {(1, 0, 0), (0, 1, 0)}. Do the vectors in this set span R³? There are lots of vectors in R³ that can be written as a linear combination of these vectors. One that can be written this way is (2, 1, 0) = 2(1, 0, 1) + 1(0, 1, 0). On the other hand there are also lots of vectors in R³ that are not linear combinations of the two vectors - there is no linear combination of (1, 0, 0) and (0, 1, 0) that produces (2, 1, 5). For this reason, the vectors (1, 0, 0) and (0, 1, 0) do not span R³.

Although the vectors of my example, (1, 0, 0) and (0, 1, 0), do not span R³, they do span a two-dimensional subspace of R³, the x-y plane.

Dafe

I think it helps to think of span just as Mark44 explains it.
In R^3 a vector such as <1, 0, 0> would span a line, in this case the x-axis. You can find any point on the x-axis by multiplying the vector with a scalar; 3<1, 0, 0> gives you a point (3,0,0) on the x-axis.

Then if you have two independent (not pointing in the same direction) vectors such as <1, 0, 0> and <0, 1, 0> they span a plane in R^3. You can call this plane a subspace of R^3, just as Mark44 writes. If you want to find a point on that plane, you take linear combinations of the vectors. Basically you move along one vector, then move along the other and add up. a<1, 0, 0> + b<0, 1, 0>.