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Finding the Orthogonal Complement

  • Thread starter Tom2
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Homework Statement



Let V be the 2-dim subspace of R^3 spanned by V1 = (1,1,1) and V2 = (-2,0,1). Find the orthogonal compliment Vperpendicular.
2. Relevant equation

X + Y + Z = 0
-5X + Y + 4Z = 0


The Attempt at a Solution


Firstly, I orthogonalize the basis for V and get the vectors (1,1,1) and (-5,1,4).

Then I apply the dot product and end up with the above equations. I found that the solution space for this system of equations is a single vector (x,y,z) = (.5z, -1.5z,z) . Usually I will get 2 vectors and then orthogonalize them as the basis for the orthogonal compliment. Can this single vector (.5,-1.5,1) be the orhogonal compliment? Or did I make a mistake somewhere? Please help.
 

Answers and Replies

  • #2
Samy_A
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Homework Helper
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Homework Statement



Let V be the 2-dim subspace of R^3 spanned by V1 = (1,1,1) and V2 = (-2,0,1). Find the orthogonal compliment Vperpendicular.
2. Relevant equation

X + Y + Z = 0
-5X + Y + 4Z = 0


The Attempt at a Solution


Firstly, I orthogonalize the basis for V and get the vectors (1,1,1) and (-5,1,4).

Then I apply the dot product and end up with the above equations. I found that the solution space for this system of equations is a single vector (x,y,z) = (.5z, -1.5z,z) . Usually I will get 2 vectors and then orthogonalize them as the basis for the orthogonal compliment. Can this single vector (.5,-1.5,1) be the orhogonal compliment? Or did I make a mistake somewhere? Please help.
(x,y,z) = (.5z, -1.5z,z) doesn't represent a single vector, but the one dimensional subspace spanned by the vector (.5,-1.5,1).
 
Last edited:
  • #3
blue_leaf77
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Firstly, I orthogonalize the basis for V and get the vectors (1,1,1) and (-5,1,4).
You don't need that. You just need to solve the system of equations
$$
x+y+z=0 \\
-2x+z=0
$$
Obviously, there will be at least one free variable, make it ##z## and solve the other variables in terms of ##z##.
 
  • #4
ehild
Homework Helper
15,426
1,823

Homework Statement



Let V be the 2-dim subspace of R^3 spanned by V1 = (1,1,1) and V2 = (-2,0,1). Find the orthogonal compliment Vperpendicular.
2. Relevant equation

X + Y + Z = 0
-5X + Y + 4Z = 0


The Attempt at a Solution


Firstly, I orthogonalize the basis for V and get the vectors (1,1,1) and (-5,1,4).

Then I apply the dot product and end up with the above equations. I found that the solution space for this system of equations is a single vector (x,y,z) = (.5z, -1.5z,z) . Usually I will get 2 vectors and then orthogonalize them as the basis for the orthogonal compliment. Can this single vector (.5,-1.5,1) be the orhogonal compliment? Or did I make a mistake somewhere? Please help.
In R3, the orthogonal complement of a two-dimensional subspace is one-dimensional, that is spanned by a single vector. You have found that vector, and the subspace consists of all vectors of form (x,y,z) = (.5z, -1.5z,z) , as you wrote. So your solution is correct, but it was not needed to orthogonalize the the basis. You can find a vector perpendicular to other two independent ones by cross-product them.
 

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