# Finding the Orthogonal Complement

• Tom2
In summary, In this problem, the goal is to find the orthogonal complement of a two-dimensional subspace in R3, which is spanned by two vectors. The orthogonal complement in R3 is a one-dimensional subspace, which is spanned by a single vector. This vector can be found by solving a system of equations, where the coefficients come from the given vectors. Alternatively, it can be found by taking the cross product of the two given vectors. In this case, the solution was found to be (x,y,z) = (.5z, -1.5z,z).
Tom2

## 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.

Tom2 said:

## 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:
Tom2 said:
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##.

Tom2 said:

## 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.

## What is the definition of the orthogonal complement?

The orthogonal complement of a vector space V is the set of all vectors in the same dimension as V that are perpendicular to every vector in V.

## How is the orthogonal complement denoted?

The orthogonal complement of a vector space V is often denoted as V.

## How do you find the orthogonal complement of a vector space?

To find the orthogonal complement of a vector space V, you can use the Gram-Schmidt process to find a basis for V. This involves choosing a basis for V and then using orthogonal projections to find a basis for V.

## What is the relationship between the orthogonal complement and the span of a vector space?

The orthogonal complement of a vector space V is the orthogonal complement of the span of V. This means that any vector in V is also perpendicular to every vector in the span of V, and vice versa.

## Why is the concept of orthogonal complement important in linear algebra?

The orthogonal complement is important because it allows us to decompose a vector space into two subspaces that are perpendicular to each other. This can be useful in many applications, including solving systems of linear equations, finding best-fit lines or planes, and understanding the geometry of vector spaces.

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