Linear span
If V is a vector space and S is a subset of V, then S spans V if every vector in V can be written as a linear combination of (finitely many) elements from S. S is then called a spanning set or generating set of V.Given any subset S of a vector space V, regardless of whether S is a spanning set of V, we can define the span of S to be the set of all linear combinations of elements of S. Then S spans V if and only if V is the span of S; in general, however, the span of S will only be a subspace of V.
A spanning set that is also linearly independent is a basis. In other words, S is a basis of V if and only if every vector in V can be written as a linear combination of elements of S in exactly one way.
Examples
The real vector space R3 has {(1,0,0), (0,1,0), (0,0,1)} as spanning set. This spanning set is actually a basis. Another spanning set for the same space is given by {(1,2,3), (0,1,2), (−1,1/2,3), (1,1,1)}, but this set isn't a basis, because it is linearly dependent. The set {(1,0,0), (0,1,0), (1,1,0)} isn't even a spanning set of R3; instead its span is the space of all vectors in R3 whose last component is zero.
thing that's done? Did you see.html">see Bartley, I'm saying to you?
MAURYA
[As before.]
Did you see Bartley?
MAURYA
[Leaves her wheel and looks out.]
God forgive you; he's riding.html">riding the mare now over the green head,
[Starts, so that her shawl falls back from her head and shows
[Coming to the fire.]
What is it ails you, at all?
MAURYA
day Bride Dara seen the dead man with the child in his arms.
CATHLEEN AND NORA
Tell us what it is you seen.
MAURYA
prayer to myself. Then Bartley came along, and he riding on
hands, as if to hide something from her eyes.] The Son of God
What is it you seen.
MAURYA
[Speaking softly.]
You did not, mother; it wasn't Michael you seen, for his body
burial by the grace of God.
MAURYA
Bartley came first on the red mare; and I tried to say "God
went by quickly; and "the blessing of God on you," says he, and
gray pony, and there was Michael upon it -- with fine clothes
[Begins to keen.]
It's destroyed we are from this day. It's destroyed, surely.
NORA
destitute with no son living?
MAURYA
will be lost now, and let you call in Eamon and make me a good
I've had a husband.html">husband, and a husband's father.html">father.html">father, and six sons in
with every one of them and they coming to the world -- and some
gone now the lot of them. . . There were Stephen, and Shawn,
Gregory of the Golden Mouth, and carried up the two of them on
something through the door that is half open behind them.]
NORA
north-east?
CATHLEEN
[Continues without hearing anything.]
There was Sheamus and his father, and his own father again,
.
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