Orthogonal Complements Prove the properties stated in Problem using the following definition, illustrated by Fig. 4. Assume that is a subspace of ℝn.
Orthogonal Complement
Let be a subspace of ℝn. A vector is orthogonal to subspace provided that is orthogonal to every vector in . The set of all vectors in ℝn that are orthogonal to is called the orthogonal complement of , denoted
.
Figure 4 An orthogonal complement to a plane
Suggested Journal Entry By Problem 70, the straight line in ℝ4 through in the direction has parametric equations
where t is a real parameter. From Problem 49, a hypeiplane in ℝ4 is the solution set of the linear equation
passing through the origin if and only if a0 = 0. But you are now in the fourth dimension, with no pictures to guide you. How do you know a line is straight? How can you tell if a hyperplane is flat? How could you define and test these concepts?
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