# Orthogonal Basis for Subspace: Find Solution

• hpayandah
In summary, the conversation is about finding an orthogonal basis for a given subspace and the solution involves using the Gram-Schmidt process. The final result is obtained by calculating (X2 . F1) / (||F1||2).
hpayandah
Hi Everyone,
I want to ask if I did this problem correctly.

## Homework Statement

Find a orthogonal basis for subspace {[x y z]T|2x-y+z=0}

## Homework Equations

X1= [3 2 -4]T, X2=[4 3 -5]T

## The Attempt at a Solution

Gram-Schmidt:
F1=X1= [3 2 -4]
F2= X2- ((X2.F1)/||F1||2)F1= [4 3 -5]T + (26/29)[3 2 -4]T

Hi hpayandah!

I don't quite get how you obtained the 26 in the end. Could you explain? (Maybe I just miscalculated it).

Hi, Thanks for replying, it's the result of:

(X2 . F1) / (||F1||2)

Attached is my work.

#### Attachments

• 20110709_015.jpg
30.3 KB · Views: 408
Yes, that is correct. You wrote X2 wrong in the OP

micromass said:
Yes, that is correct. You wrote X2 wrong in the OP
Appreciated.

## 1. What is an orthogonal basis for a subspace?

An orthogonal basis for a subspace is a set of vectors that are mutually perpendicular (orthogonal) to each other and span the entire subspace. This means that any vector in the subspace can be expressed as a linear combination of these orthogonal vectors.

## 2. How do you find an orthogonal basis for a subspace?

To find an orthogonal basis for a subspace, you can use the Gram-Schmidt process. This involves starting with a set of linearly independent vectors in the subspace and using orthogonalization to create a new set of vectors that are orthogonal to each other. This process continues until all vectors in the subspace have been accounted for.

## 3. Why is an orthogonal basis for a subspace important?

An orthogonal basis for a subspace is important because it simplifies calculations and makes it easier to solve problems involving vectors in that subspace. Additionally, it allows for easier visualization and interpretation of the relationships between vectors in the subspace.

## 4. Can an orthogonal basis for a subspace be non-unique?

Yes, an orthogonal basis for a subspace can be non-unique. This means that there can be more than one set of orthogonal vectors that span the same subspace. However, the process of finding an orthogonal basis using Gram-Schmidt will always result in the same set of vectors.

## 5. Is it possible for a subspace to not have an orthogonal basis?

No, every subspace has at least one orthogonal basis. This is because any subspace can be spanned by a set of orthogonal vectors, which can then be used to create an orthogonal basis using the Gram-Schmidt process.

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