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The example defines a subspace $W$ spanned by the first two standard basis vectors of $\mathbb{R}^4$, and its orthogonal complement $W^\perp$ spanned by the last two standard basis vectors. Together, they form a direct sum that equals the entire space $\mathbb{R}^4$.

Conditions: The ambient space is $\mathbb{R}^4$.; The bases are taken as given on the board.

The example defines a subspace $W$ spanned by the first two standard basis vectors of $\mathbb{R}^4$, and its orthogonal complement $W^\perp$ spanned by the last two standard basis vectors. Together, they form a direct sum that equals the entire space $\mathbb{R}^4$.

Conditions: The ambient space is $\mathbb{R}^4$.; The bases are taken as given on the board.