# Orthogonalizing a basis by gram schmidt process

• Kamekui
In summary, the task is to find an orthonormal basis of R^4 spanned by {1,1,1,1}, {1,0,0,1}, and {0,1,0,1}, and then use the inner product to express {2,2,2,2} as a linear combination of the basis vectors without solving the equations. The solution involves using gram schmidt orthogonalization and normalization to obtain the basis vectors, and then taking the inner product of {2,2,2,2} with each basis vector to find the coefficients for the linear combination.
Kamekui

## Homework Statement

(a.) Find an orthonormal basis of R^4 spanned by {1,1,1,1},{1,0,0,1}, and {0,1,0,1}.
(b.) Use the inner product to express {2,2,2,2} as a linear combination of the basis vectors. Do not solve the equations.

## Homework Equations

gram schmidt orthogonalization and then normalizing

## The Attempt at a Solution

(a.) I used gram schmidt orthogonalization and then normalized to get:

1/2{1,1,1,1}, 1/2{1,-1,-1,1}, 1/2{-1,1,-1,1}

(b.) I'm not sure how to do this, any help would be appreciated.

Kamekui said:

## Homework Statement

(a.) Find an orthonormal basis of R^4 spanned by {1,1,1,1},{1,0,0,1}, and {0,1,0,1}.
(b.) Use the inner product to express {2,2,2,2} as a linear combination of the basis vectors. Do not solve the equations.

## Homework Equations

gram schmidt orthogonalization and then normalizing

## The Attempt at a Solution

(a.) I used gram schmidt orthogonalization and then normalized to get:

1/2{1,1,1,1}, 1/2{1,-1,-1,1}, 1/2{-1,1,-1,1}

(b.) I'm not sure how to do this, any help would be appreciated.

So you want$$(2,2,2,2) = \frac {c_1}2 (1,1,1,1)+\frac {c_2}2 (1,-1,-1,1)+\frac {c_3}2 (-1,1,-1,1)$$What happens if you take the inner product of both sides of that with one of your basis vectors?

## 1. What is the purpose of orthogonalizing a basis?

Orthogonalizing a basis is a mathematical process used to transform a set of vectors into a new set of vectors that are orthogonal (perpendicular) to each other. This allows for easier computations and more accurate results in various applications of linear algebra.

## 2. What is the Gram Schmidt process?

The Gram Schmidt process is a method for orthogonalizing a basis by systematically constructing a new set of orthogonal vectors from a given set of linearly independent vectors. This process involves finding the projection of each vector onto the space spanned by the previously orthogonalized vectors and subtracting it from the original vector to obtain an orthogonal vector.

## 3. Why is it important to use the Gram Schmidt process?

The Gram Schmidt process is important because it allows us to transform a set of linearly independent vectors into a set of orthogonal vectors, which can simplify many mathematical calculations. It is also useful for solving systems of linear equations, finding eigenvalues and eigenvectors, and performing other operations in linear algebra.

## 4. Can the Gram Schmidt process be used for any set of vectors?

Yes, the Gram Schmidt process can be used for any set of linearly independent vectors. However, it is important to note that the resulting orthogonal vectors may not always be exact due to rounding errors or other factors. In some cases, it may be necessary to use alternative methods for orthogonalizing a basis.

## 5. What are some real-world applications of orthogonalizing a basis?

Orthogonalizing a basis has many applications in fields such as engineering, physics, and computer science. It is used in signal processing to remove noise from signals, in image processing to improve image quality, in data compression to reduce the size of data sets, and in machine learning to simplify calculations and improve accuracy. It is also used in solving systems of linear equations, finding optimal solutions in optimization problems, and in many other areas.

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