16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 2.2.9

Vector Spaces

Mathematics — Linear Algebra · direct-sum-decomposition

Problem

Let

W_1

and

W_2

be subspaces of a vector space

V

such that

W_1 + W_2 = V

and

W_1 \cap W_2 = \{0\}

. Prove that for each vector

\alpha

V

there are unique vectors

\alpha_1

W_1

and

\alpha_2

W_2

such that

\alpha = \alpha_1 + \alpha_2