Download Vector Spaces and Subspaces - 4.1 1. Vector Spaces: Definition Let

Vector Spaces and Subspaces - 4.1
1. Vector Spaces:
Definition Let V be a nonempty set of vectors, and let and be vector addition and scalar multiplication
defined on V. The set V with operations , is said be a vector space if the following
axioms must hold for all vectors u, v, and w in V, and for all scalars c and d.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
u v is in V.
u v v u.
Ÿu v w u Ÿv w .
There is a zero vector 0 in V such that u 0 u.
For each u in V, there is a vector "u in V such that u Ÿ"u 0.
cu is in V.
cŸu v cu cv.
Ÿc d u cu du.
cŸdu Ÿcd u.
1u u.
Example The set V of all nonnegative vectors in R 2 is not a vector space since "u is not in V.
Example R n with standard vector addition and scalar multiplication is a vector space.
Example Let P n be the set of polynomials of degree at most n, i.e.,
P n pŸx a 0 a 1 x T a n x n ; a Ui s are real
Define and as standard polynomial addition and scalar multiplication. Then P n with
operations and is a vector space. The zero vector here is the zero polynomial.
Example Let M n be the set of n • n matrices. Define and as standard matrix addition and scalar
multiplication. Then M n with operations and is a vector space. The zero vector here is the
n • n zero matrix.
Let M m•n be the set of m • n matrices. Define and as standard matrix addition and scalar
multiplication. Then M m•n with operations and is a vector space. The zero vector here is
the m • n zero matrix.
2. Subspaces:
Recall in Section (2.8), we defined subspaces in R n :
Definition A subspaces of R n is a set H in R n that has three properties:
a. The zero vector is in H.
b. For each u and v in H, the sum u v is in H.
c. For each u in H and each scalar c, the vector cu is also in H.
Examples are the column space of an m • n matrix is a subspace in R m and the null space of an m • n matrix
is a subspace in R n . Now we give the definition of a subspace of a space V.
1
Definition Let V with operations , three properties:
be a vector space. A subspaces of V is a set H in R n that has
a. The zero vector is in H.
b. For each u and v in H, the sum u v is in H.
c. For each u in H and each scalar c, the vector cu is also in H.
Example Let V with operations , be a vector space. Then S £0¤ and H V are subspaces of V.
Example Let H be a set of polynomials of even degrees less than 5. Let S be a set of polynomials of odd
degrees less than n. Determine if H and S are subspaces of P 4 .
pŸt a 4 t 4 a 3 t 3 a 2 t 2 a 1 t a 0 , a 4 p 0
H
qŸt b 2 t 2 b 1 t b 0 , b 2 p 0
rŸt c 0
Let q 1 Ÿt 2t 2 " t 1, and q 2 Ÿt "2t 2 " t " 3. q 1 Ÿt and q 2 Ÿt are in H. However,
q 1 Ÿt q 2 Ÿt "2t " 2 is not in H since it is linear.
So, H is not a subspace of P 4 . In a similar way, we can show that S is not a subspace of P 4 .
Example Let H be a set of even polynomials, i.e., H £pŸx a 2n x 2n a 2n"2 x 2n"2 . . . a 2 x 2 a 0 ¤. Let S
be a set of odd polynomials, i.e., S £pŸx a 2n1 x 2n1 a 2n"1 x 2n"1 . . . a 2 x¤. Determine if H
and S are subspaces of P
Check if H is a subspace of P.
a. When a 0 0, a 2 0, T , a 2n 0, pŸx 0 is in H.
b. Sum of even polynomials is again an even polynomial.
c. cpŸt is again an even polynomial.
H is a subspace of P. Since the zero vector of P is not in S, S is subspace of P.
.
Example Let v 1 , T , v k be vectors in R n . Then SpanŸv 1 , T , v k is a subspace of R n . Similarly, if V with
operations Ÿ, be a vector space, and v 1 , T , v k are vectors in V, then SpanŸv 1 , T , v k is a
subspace of V.
Example Let H be the set of all vectors of the form a " 3b, b " a, a, b and S be the set of all vectors of
the form a, b 1, 2a b, 3a " 2b where a, b are arbitrary scalars. Determine if H and S
are subspaces of R 4 .
a " 3b
Rewrite: H b"a
a
b
2
a
, S
b1
2a b
3a " 2b
.
a " 3b
b"a
Since
a
"3
1
a
b
"1
1
1
b
0
"1
, H Span
,
1
1
0
"3
1
1
0
1
0
So H is a subspace of R 4 . Since
a
1
b1
a
2a b
3a " 2b
0
2
3
0
b
1
1
0
"2
1
0
,
0
0
the zero vector
0
0
is not in S. So, S is not a subspace of R 4 .
0
Example Let H be the set of all polynomials of the form pŸt a 2t bt 2 and S be the set of all
polynomials of the form qŸt at 2 bt c. Determine if H and S are subspaces of P 2 .
H is not a subspace of P 2 since pŸt 0 is not in H. S SpanŸ1, t, t 2 .
Example Let H be the set of all n • n diagonal matrices Determine if H is a subspace of the vector space
Mn.
Check 3 properties of a subspace:
a. An n • n zero matrix is a diagonal matrix. So, the zero vector of M n is in H.
b. Let A diag a 1 T a n and B diag b 1 T b n be in H. Then A B is still in H since
A B diag a 1 b 1 T a n b n .
c. Let A diag a 1 T a n
and c be a scalar. Then cA is still in H since
cA diag ca 1 T ca n .
So, H is a subspace of M n
Example Let F be a fixed 3 • 2 matrix, and let H be the set of all matrices A in M 2•4 with the property:
FA 0 3•4 .
Determine if H is a subspace of M 2•4 .
Check 3 properties of a subspace:
a. Since F0 2•4 0 3•4 , the zero vector 0 2•4 is in H.
b. Let A and B be in H. Then FA 0 3•4 and FB 0 3•4 . Since
FŸA B FA FB 0 3•4 0 3•4 0 3•4
A B is in H.
c. Let A be in H and c be a scalar. Since
FŸcA cFA c0 3•4 0 3•4
cA is in H.
3
So, H is a subspace of M 2•4 .
Example Let H and K be subspaces of a vector space V. The intersection of H and K, written as H 9 K, is
a subset of V. Show that H 9 K is a subspace of V. The union of H and K, written as H : K is
also a subset of V. Determine if H : K is a subspace of V.
Consider H 9 K v in H and in K . Check 3 properties of a subspace:
a. Since both H and K are subspace of V, the zero vector of V is in both H and K. So, the zero vector is in
H 9 K.
b. Let u and v be in H 9 K. Then u and v are in H and K. Since H and K are subspace of V, u v is in
both H and K. So, u v is in H 9 K .
c. Let u be in H and c be a scalar. Then u is in both H and K. Since H and K are subspace of V, cu is in
both H and K. So, cu is in H 9 K.
So, H 9 K is a subspace of V.
a
Now consider H : K. Let V R 2 , H Let u 1
0
be in H and v 0
1
u v 0
, K
b
.
be in K. Then u and v are in H : K. However,
1
1
is not in H and not in K.
So, u v is not in H : K. Hence, H : K is not a subspace.
4
0