Finite dimensional counterexample for the cancellation law of the direct sum of vector spaces

Question

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?

Answer 1

Take $U$ to be the $x$ axis in the plane, $W_1$ to be the $y$ axis, and $W_2$ to be the span of the vector $(1,1)$.

Then your direct sums are both equal to the plane $\mathbb{R}^2$, but $W_1\neq W_2$.