## All length-preserving 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 length-preserving matrices are unitary and trying it out myself, I came to believe that there … Continue reading

## 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 Gram-Schmidt 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 Gram-Schmidt 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 least-squares solution to the equation , where the entries 1, , and are understood as functions of … Continue reading