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Three linearly dependent bases

Source: Simon Marais Mathematics Competition 2023 Paper B Problem 3

October 16, 2023

vectorlinear algebra

Problem Statement

Let

n

be a positive integer. Let

A,B,

and

C

be three

n

-dimensional vector subspaces of

\mathbb{R}^{2n}

with the property that

A \cap B = B \cap C = C \cap A = \{0\}

. Prove that there exist bases

\{a_1,a_2, \dots, a_n\}

of

A

,

\{b_1,b_2, \dots, b_n\}

of

B

, and

\{c_1,c_2, \dots, c_n\}

of

C

with the property that for each

i \in \{1,2, \dots, n\}

, the vectors

a_i,b_i,

and

c_i

are linearly dependent.

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