Gram-Schmidt Process: Solve for V & x

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In summary, to find the orthogonal basis, the equation v2 = w2 - (<w2,v1>)(v1)/(||v1||²) can be used. In the given example, ||v1||² was incorrectly calculated as 0 due to a miscalculation with complex numbers. The correct calculation is 2, which can then be used to obtain the orthogonal basis and compute the Fourier coefficients.
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Tonyt88
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Homework Statement



V = span(S) where S = {(1, i, 0), ((1-i), 2, 4i)}, and x = ((3+i), 4i, -4)

Obtain the orthogonal basis, then normalize for the orthonormal basis, and then compute the Fourier coefficients.

Homework Equations



v2 = w2 - (<w2,v1>)(v1)/(||v1||²)

The Attempt at a Solution



So using this above equation, I get ||v1||² to equal zero because 1 + i² = 1 - 1 = 0, thus I'm dividing by zero, so where do I go from here, or am I miscalculating somewhere?
 
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  • #2
How do you compute the square-norm of a complex number?
 
  • #3
Ok this is what you did wrong:

||v1||² = v1* . v1 = (1, i, 0)*. (1, i, 0) = 1+(-i)(i)=1+1=2
 

What is the Gram-Schmidt Process?

The Gram-Schmidt process is a mathematical method used to transform a set of linearly independent vectors into an orthogonal set of vectors. This process is commonly used in linear algebra and is named after mathematicians Jørgen Pedersen Gram and Erhard Schmidt.

What is the purpose of the Gram-Schmidt Process?

The purpose of the Gram-Schmidt process is to find an orthogonal basis for a subspace of a given vector space. This allows for easier calculations and simplifies many problems in linear algebra.

How does the Gram-Schmidt Process work?

The Gram-Schmidt process involves taking a set of linearly independent vectors and transforming them into a set of orthogonal vectors. This is done by subtracting the projections of each vector onto the previously computed orthogonal vectors. The resulting vectors are then normalized to create a set of orthonormal vectors.

What is the role of V and x in the Gram-Schmidt Process?

In the Gram-Schmidt process, V represents the set of linearly independent vectors that are being transformed into an orthogonal set. x represents the resulting orthonormal vectors after the Gram-Schmidt process has been applied. The goal is to find an orthogonal basis for V, which is achieved by the Gram-Schmidt process.

What are the applications of the Gram-Schmidt Process?

The Gram-Schmidt process has many applications in mathematics, physics, and engineering. It is commonly used in linear algebra for solving systems of linear equations, finding eigenvalues and eigenvectors, and performing matrix decompositions. It is also used in signal processing and image compression techniques.

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