Page 69 - DMTH502_LINEAR_ALGEBRA

Unit 2: Vector Subspaces

Notes
b
a b a a 1 , ,0 b b 1 , ,0
a
2
2
aa 1 ,aa 2 ,0 bb 1 ,bb 2 ,0
aa 1 bb 1 ,aa 2 bb 2 ,0 W
because aa bb ,aa bb . F
1 1 2 2
Therefore, W is a subspace of V (F).
3

Example 2: Let R be the field of real numbers. Show that

y
x ,2 ,3 : , ,z R is a subspace of V (R).
z
y
x
3
y
z
y
x
Solution: Let W x ,2 ,3 : , ,z R .
,
z
,
,
y
Let x 1 ,2 ,3z 1 , x 2 ,2y 2 ,3z be any two elements of W then x y 1 , ,x y z are
2
1
2
2
1
1
2
obviously real numbers. If a, b are two real numbers, then
a b a x 1 2y 1 3z 1 b x 2 2y 2 3z 2
ax 1 bx 2 ,2ay 1 2ay 2 ,3az 1 3az 2
,
which belongs of W ax bx ,ay by and az bz being real numbers.
1 2 1 2 1 2
Thus , R and b W
a b W .
i.e., W is subspace of V (R).
3
Example 3: If V is any vector space, V is a subspace of V; the subset consisting of the zero
vector alone is a space of V, and is called the zero subspace.
Example 4: An n × n matrix A over the field F is symmetric if A ij A ji , for each i and j. The
symmetric matrices form a subspace of the square of all n × n matrices over the field F.

Example 5: The space of polynomial functions over the field F is a subspace of the space
of all functions from F into F.

Example 6: Let F be a subfield of the field C of complex numbers, and let V be the vector
space of all 2 × 2 matrices over F. Let W be the subset of V consisting of all matrices of the form
1
a b
c 0

LOVELY PROFESSIONAL UNIVERSITY 63