Linear Algebra - Gram Schmidt & Normalization - Error in Sol

In summary, the Gram Schmidt process is used in Linear Algebra to transform a set of linearly independent vectors into a set of orthonormal vectors, while normalization is the process of scaling vectors to have a unit length. This simplifies calculations and makes it easier to solve systems of linear equations. The Gram Schmidt algorithm is a method for finding an orthonormal basis for a vector space, which involves subtracting the projection of each vector onto the previous vectors until all vectors are orthogonal and have a unit length. Normalization is important in Linear Algebra because it allows for easier comparison and manipulation of vectors, and the error in Gram Schmidt and normalization is a measure of how closely the resulting orthonormal vectors approximate the original set of vectors. This error
  • #1
YoshiMoshi
228
8

Homework Statement



So I think I found an error in the solution were it attempts to find q_2^

I'm asked find the orthornomal basis for the column space of matrix A.

equation 1.PNG


Homework Equations

The Attempt at a Solution


[/B]
My question is in what it puts for q_2^

equation 2.PNG

A_2 = [4/3 4/3 -2/3]^T
||A_2|| = sqrt((4/3)^2 + (4/3)^2 + (-2/3)^2) = 2

but in the answer key since
q_2 = A_2/||A_2|| it's implying that ||A_2|| = 3 but I calculate 2. Am I doing something wrong?

Thanks for your help.
 
Last edited:
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  • #2
Oh I see that it's correct my bad
 

Related to Linear Algebra - Gram Schmidt & Normalization - Error in Sol

1. What is the purpose of Gram Schmidt and normalization in Linear Algebra?

The Gram Schmidt process is used to transform a set of linearly independent vectors into a set of orthonormal vectors. Normalization is the process of scaling a vector to have a unit length. These techniques are important in Linear Algebra because they simplify calculations and make it easier to solve systems of linear equations.

2. What is the Gram Schmidt algorithm and how does it work?

The Gram Schmidt algorithm is a method for finding an orthonormal basis for a vector space. It involves taking a set of linearly independent vectors and transforming them into orthonormal vectors by subtracting the projection of each vector onto the previous vectors in the set. This process is repeated until all vectors in the set are orthogonal and have a unit length.

3. Why is normalization important in Linear Algebra?

Normalization is important in Linear Algebra because it allows for easier comparison and manipulation of vectors. When vectors are normalized, their lengths are all equal to 1, making it easier to determine their relative magnitude and direction. It also simplifies calculations involving dot products and projections.

4. What is the significance of the error in Gram Schmidt and normalization?

The error in Gram Schmidt and normalization is a measure of how closely the resulting orthonormal vectors approximate the original set of vectors. A small error indicates a high degree of accuracy in the transformation process, while a larger error suggests that the resulting vectors may not be as orthogonal as desired. Minimizing this error is important in order to obtain the most accurate results.

5. How can the error in Gram Schmidt and normalization be minimized?

The error in Gram Schmidt and normalization can be minimized by choosing a suitable set of initial vectors, as well as by using a more precise method for calculating the projections and subtracting them from the original vectors. It is also important to pay attention to rounding errors and use a suitable level of precision in calculations. Additionally, checking for orthogonality and unit length after the transformation can help identify and correct any errors.

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