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MAT 2401
Linear Algebra
4.3 Subspaces of Vector
Spaces
http://myhome.spu.edu/lauw
HW
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WebAssign 4.3
Written Homework
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Continue to examine the structure of
vector spaces.
Recall: Vector Spaces
Recall: Vector Spaces
Recall: Vector Spaces
Question?
Suppose V is a vector space.
If W is a subset of V, is W also a
vector space?
Subspace
A nonempty subset W of a vector space
V is called a subspace of V if W is a
vector space under the operations of
addition and scalar multiplication
defined in V.
V
W
Example 1 Subspace of R2?
W1 = {(x,y)| x-2y=0}
Example 1 Subspace of R2?
W1 = {(x,y)| x-2y=0}
Let’s check the
10 axioms!
Example 1 Subspace of R2?
W1 = {(x,y)| x-2y=0}
Hold on…check
only 4 of them….
Let’s check the
10 axioms!
Example 1 Subspace of R2?
W1 = {(x,y)| x-2y=0}
Hold on…check
only 4 of them….
Is it because
you have only
4 fingers?
Subspace
V
W
V
Subspace
V
W
May be we need
only 2…
Theorem
If W is a nonempty subset W of a
vector space V, then W is a subspace of
V if and only if
1. If u and v are in W, then u+v is in W.
2. If u is in W and c is any scalar, then
cu is in W.
Example 1 Subspace of R2?
W1 = {(x,y)| x-2y=0}
Example 1(b) Subspace of R2?
W2 = {(x,y)| x,y≥ 0}
Example 1(c) Subspace of R2?
W3 = {(0,0)}
Example 2(a) Subspace of M2,2?
S2 = the set of all 2x2 symmetric
matrices
Example 2(b) Subspace of M2,2?
W = the set of all 2x2 singular
matrices
Theorem
If V and W are both subspaces of a
vector space U. Then the intersection,
VW, is also a subspace of U.
U
W
V
0
V W