Linear Algebra Done Right · Problem 1C.23

Finite-Dimensional Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · direct-sums

Problem

Prove or give a counterexample: If

V_1, V_2, U

are subspaces of

V

such that

V = V_1 \oplus U \quad \text{and} \quad V = V_2 \oplus U,

then

V_1 = V_2

. Hint: When trying to discover whether a conjecture in linear algebra is true or false, it is often useful to start by experimenting in

\mathbf{F}^2