I am reading 'Linear Algebra and Group Lectures' written by I. Lee. By 7.5.14. of this book, the cancellation of the direct sum does NOT hold:
Let $U, W_1, W_2 \le V$ be subspaces of $V$. $U \oplus W_1 = U \oplus W_2$ does not imply $W_1 = W_2$.
The author says that only finite dimensional vector spaces will be considered in the book. However I cannot imagine a counterexample for finite $V$. Is there a 'finite' counterexample of the cancellation law of the direct sum of two vector spaces?