Vectorspace example sentences

"Vectorspace" Example Sentences

1. A vectorspace over a field F is a set V with two operations.
2. The vectorspace is characterized by the properties of the operations.
3. One example of a vectorspace is the set of polynomials with coefficients in a field.
4. A vectorspace has a zero vector that serves as an additive identity.
5. One important axiom of a vectorspace is closure under addition.
6. The vectorspace has a scalar multiplication operation that is distributive over addition.
7. The dimension of a vectorspace is the number of linearly independent vectors it contains.
8. A span of a set of vectors is the set of linear combinations that can be formed from the vectors.
9. The vectorspace of continuous functions on a closed interval is infinite-dimensional.
10. The set of all complex numbers is a vectorspace over the field of real numbers.
11. A subspace of a vectorspace is a subset that is itself a vectorspace under the same operations.
12. The kernel of a linear map is a subspace of the domain vectorspace.
13. The codomain of a linear map is a vectorspace that may be larger than the domain vectorspace.
14. The existence of a basis for a vectorspace follows from the existence of linearly independent vectors.
15. A norm on a vectorspace is a function that assigns a nonnegative value to each vector.
16. A vectorspace can have multiple norms that induce the same topology.
17. The dual space of a finite-dimensional vectorspace is isomorphic to the original vectorspace.
18. The dual space of an infinite-dimensional vectorspace is larger than the original vectorspace.
19. The set of linear functionals on a vectorspace is a vectorspace under pointwise addition and scalar multiplication.
20. The quotient of a vectorspace by a subspace is a vectorspace.
21. Two vectorspaces are isomorphic if there exists a linear transformation that bijectively maps one onto the other.
22. The eigenvectors of a linear operator on a vectorspace are the nonzero vectors that do not change direction under the operator.
23. The eigenvalues of a linear operator on a vectorspace are the scalars that multiply the eigenvectors.
24. A diagonalizable operator on a complex vectorspace admits a basis of eigenvectors.
25. A symmetric operator on a real vectorspace admits an orthonormal basis of eigenvectors.
26. The direct sum of two vectorspaces is the set of ordered pairs whose entries lie in the respective spaces.
27. The direct sum of two subspaces of a vectorspace is itself a subspace.
28. A basis of the union of two subspaces is not necessarily a union of the bases of the subspaces.
29. The rank-nullity theorem relates the dimensions of the image and kernel of a linear map between finite-dimensional vectorspaces.
30. The transpose of a linear transformation between finite-dimensional vectorspaces is represented by the transpose of the matrix that represents the transformation in some basis.
31. A bilinear form on a vectorspace is a function that takes two vectors as input and returns a scalar.
32. A quadratic form on a vectorspace is a function that takes a vector as input and returns a scalar.
33. A bilinear form is symmetric if it is unchanged upon interchanging its two arguments.
34. A bilinear form is positive definite if it takes positive values on all nonzero vectors.
35. A quadratic form is positive definite if its associated bilinear form is positive definite.
36. A projection of a vectorspace onto a subspace is a linear map that maps every vector to its nearest point in the subspace.
37. The intersection of two subspaces of a vectorspace is a subspace of the same or smaller dimension.
38. Let V be a finite-dimensional vectorspace. Then every linearly independent set of vectors in V can be extended to a basis of V.
39. A linear operator on a vectorspace is invertible if and only if its determinant is nonzero.
40. A vectorspace is called complete if every Cauchy sequence converges to a limit in the space.

Common Phrases

1. Vector space refers to a mathematical concept;
2. A vector space consists of a set of vectors;
3. These vectors can be added and multiplied by scalars;
4. The addition and scalar multiplication operations satisfy certain properties;
5. Vector spaces can have different dimensions;
6. The dimension of a vector space is defined as the number of vectors in a basis;
7. A basis is a set of linearly independent vectors that spans the vector space;
8. Vector spaces are important in many areas of mathematics and physics;
9. They are used in linear algebra, functional analysis, and differential geometry;
10. Vector spaces also have applications in computer science and engineering.