Direct sum and subspaces

Question

The question states, "Find subspaces W, X, Y ⊂ ℝ2 with ℝ2=W⊕X=W⊕Y, but X does not equal Y." So if W and X are the Axis in R2 and Y can not equal X, how do you get W and Y to equal R2 without Y equaling X?

Answer 1

$W\oplus X = \Bbb{R}^2$ does not necessarily mean that $W$ and $X$ are the coordinate axes, it just means that any vector in $\Bbb{R}^2$ can be written as as a linear combination of vectors in $X$ and vectors in $Y$ (or, equivalently, as a sum of a vector from $X$ and a vector from $Y$).

To answer your question the fact that $\Bbb{R}^2$ is $2$-dimensional can be useful: since the dimension of a direct sum is the sum of the dimensions of the (distinct) summands, you can actually pick $\textit{any}$ three $1$-dimensional (i.e. nontrivial and proper) subspaces of $\Bbb{R}^2$.

Once you choose three subspaces $W$, $X$, and $Y$, think about how you can write any vector in $\Bbb{R}^2$ as a sum of an element of $W$ and one of $X$, or as a sum of an element of $W$ and one of $Y$.