Vector Spaces
& Subspaces
Kristi Schmit
Definitions
• A subset W of vector space V is called a subspace of
V iff
a.
The zero vector of V is in W.
b.
W is closed under vector addition, for each u and v in
W, the sum u + v is in W.
W is closed under multiplication by scalars, for each u
in W and each scalar c, the vector cu is in W.
c.
•
Any subspace W of vector space V is a vector
space.
Example 1
Let W be the set of all vectors of the form shown,
where a and b represent arbitrary real numbers. In
each case, either find a set S of vectors that spans W or
give an example to show that W is not a vector space.
é 2a + 3b
ê
W = ê -1
ê 2a - 5b
ë
ù
ú
ú
ú
û
Response
é 2a + 3b
ê
W = ê -1
ê 2a - 5b
ë
•
•
•
ù
ú
ú
ú
û
The zero vector of V is not in W because of
the -1 in the subset.
Therefore the subset fails the first property of
a subspace.
Thus, W is not a subspace of V and therefore
is not a vector space.
Definitions
If v1 ,…., vp are in an n-dimensional vector space over the real
numbers, Rn, then the set of all linear combinations of v1 ,…., vp
is denoted by Span{v1 ,…., vp } and is called the subset of Rn
spanned by v1 ,…., vp. That is, Span{v1 ,…., vp } is the collection
of all vectors that can be written in the form
c1v1+ c2v2+ …+ cpvp
with c1,…, cp scalars.
Example 2
Let W be the set of all vectors of the form shown,
where a and b represent arbitrary real numbers. In
each case, either find a set S of vectors that spans W or
give an example to show that W is not a vector space.
é 2a - b ù
ê
ú
3b - c ú
ê
W=
ê 3c - a ú
ê 3b ú
ë
û
Response
é 2a - b ù é
ê
ú ê
3b
c
ú=ê
W =ê
ê 3c - a ú ê
ê 3b ú ê
ë
û ë
ì é
ï ê
ï ê
ía
ï ê
ïî êë
2
0
-1
0
ù é
ú ê
ú + bê
ú ê
ú ê
û ë
-1
3
0
3
-1
3
0
3
2
0
-1
0
ù é
ú ê
ú + cê
ú ê
ú ê
û ë
0
-1
3
0
0
-1
3
0
ù
ú
ú
ú
ú
û
ü
ù
ï
ú
ú : a, b, c Î Rïý
ú
ï
ú
ïþ
û
the set of all linear combinations of v1 ,…., vp
ìé
ïê
ï
Span = íê
ïê
ïîêë
2
0
-1
0
ùé
úê
úê
úê
úê
ûë
-1
3
0
3
ùé
úê
úê
úê
úê
ûë
0
-1
3
0
ùü
úïï
úý
úï
úï
ûþ