Further, it is established that a locally convex topological vector space is Volterra if and only if it is Baire; and the weak topology of a topological vector space fails to be Baire if the dual of the space contains an infinite linearly independent pointwise bounded subset....
When is a matrix P invertible, and how to find it? For an invertible matrix A if A^ {-1} = A^T, prove that det (A) = +1 and -1. When does a matrix have linearly independent eigenvectors? How do you determine if a square matrix is invertible?
Since then the study of matroids has blossomed into a large and beautiful theory, one part of which is the characterization of the greedy algorithm: greedy is optimal on a problem if and only if the problem can be represented as a matroid. Mathematicians have also characterized which matroids ...
When inspecting the figures in a vector graphics program such as Inkscape or Illustrator, the font size is correct when the height is smaller than a certain threshold, but at larger heights, the actual font size increases linearly. The exact slope and onset appears to be system dependent, an...
"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." - Paul Halmos
Linear Transformations on a Vector SpaceLet T be a linear transformation from vector space V to vector space W. Then we prove or disprove, using a counterexample, whether the zero vector in V always gets mapped to the zero vector in W v...
Semi-inner products are, in general, non-unique: a normed space X has a unique semi-inner product if and only if it is smooth, that is every non-zero vector x\in X admits a unique norming functional, that is, a norm-one functional \varphi _x\in X^* such that \langle \varphi _...
and an imaginary line from the core of the ball through the center of the dot and outward as a vector, then you can picture aBloch spherelike this one. Each superposition state a qubit may take may be represented by a vector in a Bloch sphere, which you can think of i...
Such problems are characterized by an under-determined linear system (1)y=Axwhere y∈Z≥0n is an observed vector of counts, x∈Z≥0r is a vector of latent counts of interest with r>n, and A is an n×r matrix with non-negative integer entries. This kind of structure occurs commonly...
The matrix can be of the form m×n also it is known that the nullspace lies inside the domain, and the column space lies inside the codomain. For the matrix we have the null space equals column space.Answer and Explanation: Become a member and unlock all Study Answers Start today. Try...