Question 1160968: Let L be the line spanned by [-1, 4, 9, 0] in R^4
Find a basis of the orthogonal complement L⊥ of L.
Answer by ikleyn(52781)   (Show Source):
You can put this solution on YOUR website! .
Let L be the line spanned by [-1, 4, 9, 0] in R^4.
Find a basis of the orthogonal complement L⊥ of L.
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The orthogonal complement to the given line L is the set of all vectors (x1,x2,x3,x4) in R^4
such that
-x1 + 4x2 + 9x3 = 0. (1)
Looking at this equation, we can guess mentally its 3 linearly independent vector solutions
v1 = (4,1,0,0),
v2 = (9,0,1,0),
v3 = (0,-9,4,1).
It is clear and easy to check that these vectors satisfy equation (1), so they all belong
to the orthogonal complement L⊥ of L.
It is also easy to check that vectors v1, v2 and v3 are linearly independent,
so they form a basis in the orthogonal complement.
At this point, the problem is solved completely.
The goal of this problem is to help to a student to develop an intuition,
necessary for solving elementary tasks in linear algebra.
To have this intuition is the same as to keep the multiplication table
in your mind: without it, there is no way in the subject.
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