Orthogonal basis to two vectors in R4

In summary, we are looking for vectors that are perpendicular to (1,4,4,1) and (2,9,8,2) in four-dimensional space. Using Gauss elimination, we can reduce the problem to finding a basis of four vectors. This is because the orthogonal subspace is also two-dimensional. To find these vectors, we set up two equations in four unknowns and solve for two free variables. This results in two additional vectors that can be added to the original two to form a basis.
  • #1
jayred
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0

Homework Statement



Find all vectors that are perpendicular to (1,4,4,1) and (2,9,8,2)

The Attempt at a Solution



Create matrix A = [[1,4,4,1],[2,9,8,2]]
Set Ax = 0
Reduce by Gauss elimination
Produces basis of (-4,0,1,0) and (-1,0,0,1)

I don't know what the correct solution to this problem is, but as far as I understand it, it would seem that the basis should be one dimensional as the two given vectors form a plane and only a line is orthogonal to a plane, not a plane.
 
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  • #2
You are in FOUR dimensions. The original vectors span a 2 dimensional subspace. The orthogonal subspace ought to also be 2 dimensional. Only in three dimensions would the space orthogonal to a two dimensional space be a line.
 
  • #3
But these are vectors in R^4, so your basis will have to have four vectors. True, the given vectors form a plane, but it's a plane in four-dimensional space. There are two more dimensions that aren't in this plane.

Start with the vector (x, y, z, w). It has to be perpendicular to (1, 4, 4, 1) and (2, 9, 8, 2), so find the dot the first vector with each of the two others. That will give you two equations in four unknowns, so there will be two variables that are free. Choose convenient values to get two more vectors for your basis.
 

1. What is an orthogonal basis in R4?

An orthogonal basis in R4 refers to a set of four vectors that are mutually perpendicular, meaning they are at right angles to each other. This type of basis is useful for representing and performing calculations in four-dimensional space.

2. How do you determine if two vectors in R4 are orthogonal?

To determine if two vectors in R4 are orthogonal, you can use the dot product. If the dot product of the two vectors is equal to zero, then they are orthogonal. This means that the vectors are perpendicular and form a right angle.

3. Can two vectors in R4 be orthogonal but not form a basis?

Yes, two vectors in R4 can be orthogonal but not form a basis. In order for a set of vectors to form a basis, they must be linearly independent and span the entire space. However, two orthogonal vectors in R4 may not be linearly independent, meaning they are not enough to span the entire four-dimensional space.

4. How do you find an orthogonal basis for a set of vectors in R4?

To find an orthogonal basis for a set of vectors in R4, you can use the Gram-Schmidt process. This involves taking the original set of vectors and using a series of orthogonalization and normalization steps to transform them into a set of mutually orthogonal vectors.

5. Is an orthogonal basis unique for a set of vectors in R4?

Yes, an orthogonal basis is unique for a set of vectors in R4. This means that there is only one set of mutually orthogonal vectors that can span the four-dimensional space. However, there may be multiple ways to arrive at this unique basis using different methods, such as the Gram-Schmidt process.

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