Section 4.3: SUBSPACES OF VECTOR SPACES
When you are done with your homework you should be able to…
 Determine whether a subset W of a vector space V is a subspace of V
 Determine subspaces of R n
SUBSPACES
In many applications of linear algebra, vector spaces occur as a _____________
of larger spaces. A _________________ subset of a vector ____________ is a
________________ when it is a vector _________________ with the
___________ operations defined in the _______________ vector space.
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Consider the following: W   0, y  and V  R .
DEFINITION OF A SUBSPACE OF A VECTOR SPACE
A nonempty subset W of a vector space V is called a __________________ of
V when _______ is a vector space under the operations of ________________
and _______________ ____________________________ defined in V .
THEOREM 4.5: TEST FOR A SUBSPACE
If W is a nonempty subset of a vector space V , then W is a subspace of V if and
and only if the following closure conditions hold.
1. If u and v are in W , then _____________ is in W .
2. If u is in W and c is any scalar, then __________ is in W .
Example 1: Verify that W is a subspace of V .
a. W   x, y, 2 x  3 y  : x and y  
V  R3
b. W is the set of all functions that are differentiable on  1,1 . V is the set
of all functions that are continuous on  1,1 .
Example 2: Verify that W is not a subspace of the vector space by giving a
specific example that violates the test for vector subspace.
a. W is the set of all linear functions ax  b , a  0 in C   ,   .
b. W is the set of all matrices in M 3,1 , of the form  a
T
0
a  .
THEOREM 4.6: THE INTERSECTION OF TWO SUBSPACES IS A SUBSPACE
If V and W are both subspaces of a vector space U , then the intersection of V
and W , denoted by _____________________, is also a subspace of U .
Example 3: Determine whether the subset C   ,   is a subspace of C   ,   .
a. The set of all negative functions:
b. The set of all odd functions:
f  x  0 .
f x   f  x .
Example 4: Determine whether the subset of M n , n is a subspace of M n , n with the
standard operations of matrix addition and scalar multiplication.
a. The set of all negative functions:
b. The set of all odd functions:
f  x  0 .
f x   f  x .
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Example 5: Determine whether the set W is a subspace of R with the standard
operations. Justify you’re your answer.
W   x1 , x2 , 4  : x1 and x2   