How to check if a vector is an eigenvector of a matrix? How to check if a matrix is a basis? Is the determinant of a matrix always positive? Prove that if A is a m x n matrix and B is a nxm matrix where m > n, then det(AB) = 0 . ...
Schmidt process to get an orthonormal basis {eq}B {/eq} for {eq}S {/eq} starting with any basis for {eq}S {/eq}, or even a set of generators for {eq}S. {/eq} An orthonormal basis for {eq}S {/eq} provides a quick tool to construct the orthogonal projection of a vector.Ans...
This assertion allows one to prove that: 1) if d n > 0, , then there exist an orthonormal basis { n } v1 and complete but not hereditarily complete biorthogonal families \\\(\\\mathfrak{X}, \\\mathfrak{X}'\\\) in H, such that ∥ X n - n ∥d n , ∥x′ n - n ∥ ...
In order for f to be well-defined, we must have \sum_{i=1}^\infty |f(e_i)| < \infty . Proof: Follows directly from definition of basis and linearity of f . \square Definition 1 We say that x_n \to x , or that x_n converges to x strongly with respect to some norm \l...
How to normalize vectors to produce an orthonormal set? How to prove the basis of a vector space? What is the basis of a vector space? The orthogonal projection of b as: orth_ab = b - proj_ab Geometrically this is a projection of b onto a vector perpendicular to a. Consider a = ...
How to prove that a subspace is the span of vectors? How to find the dimension of a subspace from nullspace? Is this set a subspace of \mathbb{R}^3 or not? If so, explain why. Span: how to find the basis of a subspace How to find basis of subspace? How do you find the basi...
How to determine if a set of vectors form a basis? How to check if a vector is in a plane? How to show vectors are linearly dependent? Suppose v_{1}, v_{2}, v_{3} are vectors in R_{n}, and that u and w are both in Span{ v_{1}, v_{2}, v_{3. Prove that for ...
To prove the spherical symmetry of intra-atomic SCF in general, we need to evaluate minimum radius of the electron orbits in atom and compare it with maximum electric charge radius of the corresponding atomic nucleus. Since electrons in low-lying orbits in heavy atoms move with relativistic ...
To prove the spherical symmetry of intra-atomic SCF in general, we need to evaluate minimum radius of the electron orbits in atom and compare it with maximum electric charge radius of the corresponding atomic nucleus. Since electrons in low-lying orbits in heavy atoms move with relativistic ...
Additionally, this new approach may prove to be useful to those interested in a quantized theory of space-time, as we believe this requires a quantized measure for the quantification of the independent degrees of freedom. Keywords: quantum mechanics; measure theory; non-additive measures; ...