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I want to start off by saying thank you so much for reading my question. The question goes as following:

Give an example of a vector space V and non-trivial subspaces $X, Y, Z$ of V such that $V = X \oplus Y = X \oplus Z$ but $Y$ is not equal to $Z$. (Hint: You can find examples in R^2)

My attempt was to use complex numbers but I think they want me to be in $\mathbb{R}^2$. I thought of making $X=(x,0)$ and $Y=(0,y)$. I just want to know if I'm on the right track.

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    And what options you have in mind for $Z$? – 2017-01-29
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    Hint: Any basis for $\mathbb{R}^2$ needs just two elements. Any two distinct lines through the origin in this space will span it. Try a few combinations to see how this can lead to a direct sum. – 2017-01-29
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    And yes, you are on the right track. Complex entries are not needed. – 2017-01-29
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    I appreciate your input thank you so much. – 2017-01-29

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If we choose $$V=\mathbb{R}^2,\;X=\text{Span }\{(1,0)\},\;Y=\text{Span }\{(0,1)\},\;Z=\text{Span }\{(1,1)\}$$ then, $V=X\oplus Y=X\oplus Z\text{ and }Y\ne Z.$

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    You are welcome! – 2017-01-29