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Kunze Linear Algebra
· Problem 2.2.9
Vector Spaces
Mathematics — Methods · direct-sum-decomposition
Problem
Let
W
1
W_1
W
1
and
W
2
W_2
W
2
be subspaces of a vector space
V
V
V
such that
W
1
+
W
2
=
V
W_1 + W_2 = V
W
1
+
W
2
=
V
and
W
1
∩
W
2
=
{
0
}
W_1 \cap W_2 = \{0\}
W
1
∩
W
2
=
{
0
}
. Prove that for each vector
α
\alpha
α
in
V
V
V
there are unique vectors
α
1
\alpha_1
α
1
in
W
1
W_1
W
1
and
α
2
\alpha_2
α
2
in
W
2
W_2
W
2
such that
α
=
α
1
+
α
2
\alpha = \alpha_1 + \alpha_2
α
=
α
1
+
α
2
.
Related theory
Direct Sum Of Eigenspaces
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