Prove Theorem 5.1.8d. for any subspace V of .
It is proved that .
Consider a subspace V of such that and .
If the vectoris perpendicular then .
By the definition, the dot product of the vectors and is defined as follows.
Similarly, for role="math" localid="1659438267836" and , the dot product of the vectors and is defined as follows.
Therefore, the set V is subset of .
Theorem: Property of the orthogonal compliment.
Consider a subspace V of .
a. The orthogonal complement role="math" localid="1659438558378" of is a subspace of .
b. The intersection of V and consist of the zero vector: .
By the theorem, the equation for the set V is defined as follows.
By the theorem, the equation for the set is defined as follows.
Compare the equations and as follows.
As and implies .
Hence, the values for subset V of .
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