16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 6.6.11

Elementary Canonical Forms

Mathematics — Linear Algebra · dual-space-direct-sum

Problem

Let

V

be a vector space, let

W_1, \ldots, W_k

be subspaces of

V

, and let

V_j = W_1 + \cdots + W_{j-1} + W_{j+1} + \cdots + W_k.

Suppose that

V = W_1 \oplus \cdots \oplus W_k

. Prove that the dual space

V^*

has the direct-sum decomposition

V^* = V_1^0 \oplus \cdots \oplus V_k^0