Find an orthogonal basis for the subspace of

Thread starter Jamin2112
Start date Aug 2, 2011
Tags

Basis Orthogonal Subspace

In summary, an orthogonal basis is a set of vectors that are mutually perpendicular and have a length of 1. It is important because it simplifies vector operations and calculations, provides a geometric interpretation of vector spaces, and can be used to find solutions to systems of equations. An orthogonal basis can be found using the Gram-Schmidt process or the QR decomposition of a matrix. The purpose of finding an orthogonal basis for a subspace is to simplify calculations and operations within that subspace, and it can be found for any subspace but may not always consist of unit vectors.

Aug 2, 2011

Jamin2112

Homework Statement

... R⁴ consisting of all vectors of the form [a+b a c b+c]

Homework Equations

Gram-Schmidt process, perhaps?

The Attempt at a Solution

Not sure how to approach this one. Helpful hint?

Physics news on Phys.org

Aug 2, 2011

Jamin2112

Never mind! I think i figured it out!

Related to Find an orthogonal basis for the subspace of

1. What is an orthogonal basis?

An orthogonal basis is a set of vectors that are mutually perpendicular (orthogonal) to each other and have a length of 1 (unit length). This means that the dot product of any two vectors in the basis is equal to 0, and each vector in the basis is independent from the others.

2. Why is an orthogonal basis important?

An orthogonal basis is important because it simplifies vector operations and calculations. It also helps to avoid errors and confusion when working with vectors. Additionally, an orthogonal basis can provide a geometric interpretation of vector spaces and can be used to easily find solutions to systems of equations.

3. How do you find an orthogonal basis?

To find an orthogonal basis, you can use the Gram-Schmidt process. This involves starting with a set of linearly independent vectors and using orthogonal projections and normalization to create a set of orthogonal vectors. Another method is to use the QR decomposition of a matrix to find an orthogonal basis for the column space of the matrix.

4. What is the purpose of finding an orthogonal basis for a subspace?

The purpose of finding an orthogonal basis for a subspace is to simplify calculations and operations within that subspace. It also allows for a geometric interpretation of the subspace and can make it easier to find solutions to problems within that subspace.

5. Can an orthogonal basis be found for any subspace?

Yes, an orthogonal basis can be found for any subspace. However, in some cases, it may not be possible to find an orthogonal basis that consists of only unit vectors. In these cases, a normalized orthogonal basis can still be found, but the vectors may not have a length of 1.