16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 6.6.2

Elementary Canonical Forms

Mathematics — Linear Algebra · direct-sum-dimension

Problem

Let VV be a finite-dimensional vector space and let W1,…,WkW_1, \ldots, W_k be subspaces of VV such that V=W1+⋯+Wkanddim⁡V=dim⁡W1+⋯+dim⁡Wk.V = W_1 + \cdots + W_k \quad \text{and} \quad \dim V = \dim W_1 + \cdots + \dim W_k. Prove that V=W1⊕⋯⊕WkV = W_1 \oplus \cdots \oplus W_k.

Related theory

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