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A general vector in $\mathbb{R}^4$ is decomposed by separating its coordinates into two blocks. The first two coordinates form a vector in $W$, and the last two coordinates form a vector in $W^\perp$.

Conditions: The ambient space is $\mathbb{R}^4$.; $W$ has basis $\left\{\begin{pmatrix}1\\0\\0\\0\end{pmatrix},\begin{pmatrix}0\\1\\0\\0\end{pmatrix}\right\}$.; $W^\perp$ has basis $\left\{\begin{pmatrix}0\\0\\1\\0\end{pmatrix},\begin{pmatrix}0\\0\\0\\1\end{pmatrix}\right\}$.; The vector is represented as $\begin{pmatrix}a\\b\\c\\d\end{pmatrix}$ with $a,b,c,d \in \mathbb{R}$.

A general vector in $\mathbb{R}^4$ is decomposed by separating its coordinates into two blocks. The first two coordinates form a vector in $W$, and the last two coordinates form a vector in $W^\perp$.

Conditions: The ambient space is $\mathbb{R}^4$.; $W$ has basis $\left\{\begin{pmatrix}1\\0\\0\\0\end{pmatrix},\begin{pmatrix}0\\1\\0\\0\end{pmatrix}\right\}$.; $W^\perp$ has basis $\left\{\begin{pmatrix}0\\0\\1\\0\end{pmatrix},\begin{pmatrix}0\\0\\0\\1\end{pmatrix}\right\}$.; The vector is represented as $\begin{pmatrix}a\\b\\c\\d\end{pmatrix}$ with $a,b,c,d \in \mathbb{R}$.

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.

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.