Direct Sum of vector subspaces

Solution 1:

For instance, in $\mathbb{R}^3$

$$ \langle e_1 \rangle + \langle e_2, e_3 \rangle = \langle e_1 \rangle \oplus \langle e_2, e_3 \rangle $$

is a direct sum. Whereas

$$ \langle e_1, e_2 \rangle + \langle e_2, e_3 \rangle $$

is not.

EDIT. $\langle e_1 \rangle $ stands for the vector subspace generated by vector $e_1 = (1,0,0)$. Maybe you write it $\mathrm{span}(e_1)$? The output of both sums is the same, namely the whole $\mathbb{R}^3$.

The point is that the first sum is a direct one because $\langle e_1 \rangle \cap \langle e_2, e_3\rangle = \left\{ (0,0,0)\right\}$, whereas $\langle e_1,e_2 \rangle \cap \langle e_2, e_3 \rangle = \langle e_2\rangle$. Hence the last one is not a direct sum.