Page 269 - 35Linear Algebra

14.6 Orthogonal Complements 269

There is not a unique answer to this question as can be seen from the following
3
picture of subspaces in W = R .

However, using the inner product, there is a natural candidate U ⊥ for this
second subspace as shown below.

Definition If U is a subspace of the vector space W then the vector space

⊥

U := w ∈ W w u = 0 for all u ∈ U

is the orthogonal complement of U in W.
⊥
Remark The symbols “U ” are often read as “U-perp”. This is the set of all vectors
in W orthogonal to every vector in U. Notice also that in the above definition we
have implicitly assumed that the inner product is the dot product. For a general inner

product, the above definition would read U ⊥ := w ∈ W hw, ui = 0 for all u ∈ U .

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