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Author Archives: hecker
All lengthpreserving matrices are unitary
I recently read the (excellent) online resource Quantum Computing for the Very Curious by Andy Matuschak and Michael Nielsen. Upon reading the proof that all lengthpreserving matrices are unitary and trying it out myself, I came to believe that there … Continue reading
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Linear Algebra and Its Applications, Exercise 3.4.28
Exercise 3.4.28. Given the plane and the following vectors in the plane, find an orthonormal basis for the subspace represented by the plane. Report the dimension of the subspace and the number of nonzero vectors produced by GramSchmidt orthogonalization. Answer: … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.27
Exercise 3.4.27. Given the subspace spanned by the three vectors find vectors , , and that form an orthonormal basis for the subspace. Answer: We can save some time by noting that and are already orthogonal. We can normalize these … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.26
Exercise 3.4.26. In the GramSchmidt orthogonalization process the third component is computed as . Verify that is orthogonal to both and . Answer: Taking the dot product of and we have Since and are scalars and and are orthonormal we … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.25
Exercise 3.4.25. Given over the interval what is the closest line to the parabola formed by ? Answer: This amounts to finding a leastsquares solution to the equation , where the entries 1, , and are understood as functions of … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.24
Exercise 3.4.24. As discussed on page 178, the first three Legendre polynomials are 1, , and . Find the next Legendre polynomial; it will be a cubic polynomial defined for and will be orthogonal to the first three Legendre polynomials. … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.23
Exercise 3.4.23. Given the step function with for and for , find the following Fourier coefficients: Answer: For the numerator is and the denominator is so that . For the numerator is so that . For the numerator is and … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.22
Exercise 3.4.22. Given an arbitrary function find the coefficient that minimizes the quantity (Use the method of setting the derivative to zero.) How does this value of compare with the Fourier coefficient ? What is if ? Answer: We are … Continue reading
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Linear Algebra and Its Applications, Exercise 3.4.21
Exercise 3.4.21. Given the function on the interval , what is the closest function to ? What is the closest line to ? Answer: To find the closest function to the function we first project onto the function on the … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.20
Exercise 3.4.20. Given the vector what is the length ? Given the function for what is the length of the function over the interval? Given the function for what is the inner product of and ? Answer: We have so … Continue reading