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The lecturer states that the representation is unique because, in the context of a direct sum $W \oplus W^\perp = \mathbb{R}^4$, there is only one way to split a given vector into a part belonging to $W$ and a part belonging to $W^\perp$. No other pair of vectors from these subspaces can sum to the same original vector.

Conditions: The ambient space is $\mathbb{R}^4$.; $W$ and $W^\perp$ are the specific subspaces defined on the board.; The equation $W \oplus W^\perp = \mathbb{R}^4$ holds.

The lecturer states that the representation is unique because, in the context of a direct sum $W \oplus W^\perp = \mathbb{R}^4$, there is only one way to split a given vector into a part belonging to $W$ and a part belonging to $W^\perp$. No other pair of vectors from these subspaces can sum to the same original vector.

Conditions: The ambient space is $\mathbb{R}^4$.; $W$ and $W^\perp$ are the specific subspaces defined on the board.; The equation $W \oplus W^\perp = \mathbb{R}^4$ holds.