OrthogonalComplement.md (943B)
1 # Orthogonal Complement 2 3 Khan U3 4 5 **Definition:** The orthogonal complement of a subspace is the subspace such that the dot product between any vectors (one from each subspace) are 0. 6 7 The orthogonal complement of the subspace V in $\R^n$ is defined as follows: 8 9 $V^\perp = \{\vec{x} \in \R^n | \vec{x} \cdot \vec{v} = 0 \text{ and } \vec{v} \in V \}$ 10 11 The orthogonal complement of a subspace is a subspace in all cases as it respects scalar multiplication, vector addition, and contains the zero vector. 12 13 Every element of the nullspace is in the orthogonal complement and vice versa thus they are the same set. 14 15 ## Dimensionality 16 17 For the arbitrary subspace V, we know dim(V) = k. As such, we also know for O which is the orthogonal complement, that dim(O) = k - n where R^n is the [AmbientSpace](AmbientSpace.md) 18 19 This is given because we also know that the [Nullity](Nullity.md) + [Rank](Rank.md) = dim([Ambient Space](AmbientSpace.md)).