선형 변환

노트

위키데이터

• ID : Q207643

말뭉치

1. Is there an intuitive reason why the first definition is called a linear map, and why you would not call y=1+x a linear map, despite the fact that it defines a straight line on a plane?
2. The composition of two or more linear maps (also called linear functions or linear transformations) enjoys the same linearity property enjoyed by the two maps being composed.
3. Proposition Let , and be three linear spaces.
4. Then, where: in step we have used the fact that is linear; in step we have used the linearity of .
5. In a previous lecture, we have proved that matrix multiplication defines linear maps on spaces of column vectors.
6. If we start with the linear map \(T \), then the matrix \(M(T)=A=(a_{ij})\) is defined via Equation 6.6.1.
7. Recall that the set of linear maps \(\mathcal{L}(V,W) \) is a vector space.
8. Since we have a one-to-one correspondence between linear maps and matrices, we can also make the set of matrices \(\mathbb{F}^{m\times n} \) into a vector space.
9. Next, we show that the composition of linear maps imposes a product on matrices, also called matrix multiplication.
10. However, we still require the domain of the partial function to be a linear subspace, after which the definition above applies.
11. Notice that we do not require partially-defined linear operators to be continuous; see unbounded operator.
12. Learning Objectives Describe the kernel and image of a linear transformation.
13. Here we consider the case where the linear map is not necessarily an isomorphism.
14. Definition Kernel and Image Let \(V\) and \(W\) be vector spaces and let \(T:V\rightarrow W\) be a linear transformation.
15. : Kernel and Image as Subspaces Let \(V,W\) be vector spaces and let \(T:V\rightarrow W\) be a linear transformation.
16. You now know what a transformation is, so let's introduce a special kind of transformation called a linear transformation.
17. It only makes sense that we have something called a linear transformation because we're studying linear algebra.
18. We already had linear combinations so we might as well have a linear transformation.
19. And a linear transformation, by definition, is a transformation-- which we know is just a function.
20. Thus, a linear map is said to be operation preserving.
21. It is then necessary to specify which of these ground fields is being used in the definition of "linear".
22. If V and W are spaces over the same field K as above, then we talk about K-linear maps.
23. is a real matrix, then defines a linear map from ℝ to ℝ by sending the column vector to the column vector .
24. The goal of this paper is to review some work on agent-based financial market models in which the dynamics is driven by piecewise-linear maps.
25. The goal of our paper is to review agent-based financial market models in which the dynamics is driven by piecewise-linear maps.
26. One important advantage of piecewise-linear maps is that they allow very clear analytical insights into the functioning of the underlying model.
27. Unfortunately, piecewise-linear models are still not completely understood.
28. So to finish, we need only check that h {\displaystyle h} is linear.
29. So not only is any linear map described by a matrix but any matrix describes a linear map.
30. With the theorem, we have characterized linear maps as those maps that act in this matrix way.
31. Each linear map is described by a matrix and each matrix describes a linear map.
32. The examples of linear mappings from that we introduced in Section ?? were matrix mappings.
33. that linear mappings of to are determined by their values on the standard basis .
34. Use the dot product to define the mapping by Then is linear.
35. More precisely, if then Linear independence implies that ; that is .
36. The main example of a linear transformation is given by matrix multiplication.
37. When and are finite dimensional, a general linear transformation can be written as a matrix multiplication only after specifying a vector basis for and .
38. A good deal of mathematics is devoted to reducing questions concerning arbitrary mappings to linear mappings.
39. On the other hand, it is often possible to approximate an arbitrary mapping by a linear one, whose study is much easier than the study of the original mapping.
40. There are many characterizations of linear operators from various matrix spaces into themselves which preserve term rank.
41. That is, a linear mapfrommatrix spaces intomatrix spaces preserves any two term ranks if and only ifpreserves all term ranks if and only ifis a ()-block map.
42. This story shows how to identify a linear transformation based only on the transformation of a few points.
43. For the math in this story, I used this work of Alexander Nita, and descriptions from Wikipedia (linear map).
44. If V and W are finite-dimensional vector spaces and a basis is defined for each vector space, the linear map f can be represented by a transformation matrix.
45. Therefore, if we have a vector v, a basis in both vector space(V, W) and m points with {v, f(v)} pair we can determine linear transformation.
46. Of most interest would be any automated processing of OpenStreetMap data into a linear map.
47. Overpass API#Public transport example - Overpass API can produce linear maps of transport network data from OpenStreetMap.
48. Our analysis reveals that a simple law governing cell-size control—a noisy linear map—explains the origins of these cell-size oscillations across all strains.
49. The image is divided into blocks, and each block is transformed using a linear mapping.
50. A linear transformation is a function from one vector space to another that respects the underlying (linear) structure of each vector space.
51. A linear transformation is also known as a linear operator or map.
52. Linear transformations are useful because they preserve the structure of a vector space.
53. So, many qualitative assessments of a vector space that is the domain of a linear transformation may, under certain conditions, automatically hold in the image of the linear transformation.