Geometric Operations
... It is possible to rotate all pixels onto negative coordinate values and hence to produce an empty destination. Example: If an image is rotated by 180 degrees, all pixels except the origin fall outside of the bounds of the source. In this case the AffineTransformOp actually gives rise to an exception ...
... It is possible to rotate all pixels onto negative coordinate values and hence to produce an empty destination. Example: If an image is rotated by 180 degrees, all pixels except the origin fall outside of the bounds of the source. In this case the AffineTransformOp actually gives rise to an exception ...
MATH 412: NOTE ON INFINITE-DIMENSIONAL VECTOR SPACES
... we choose the answer to be “yes”. 2. General definitions We now summarize what we saw in the examples above. For this let V be a vector space over a field F , and fix S ⊂ V . The proofs are all the same as in the finite-dimensional case and can also be found in my notes for Math 223 (see my website) ...
... we choose the answer to be “yes”. 2. General definitions We now summarize what we saw in the examples above. For this let V be a vector space over a field F , and fix S ⊂ V . The proofs are all the same as in the finite-dimensional case and can also be found in my notes for Math 223 (see my website) ...
MATH 412: NOTE ON INFINITE-DIMENSIONAL VECTOR SPACES
... we choose the answer to be “yes”. 2. General definitions We now summarize what we saw in the examples above. For this let V be a vector space over a field F , and fix S ⊂ V . The proofs are all the same as in the finite-dimensional case and can also be found in my notes for Math 223 (see my website) ...
... we choose the answer to be “yes”. 2. General definitions We now summarize what we saw in the examples above. For this let V be a vector space over a field F , and fix S ⊂ V . The proofs are all the same as in the finite-dimensional case and can also be found in my notes for Math 223 (see my website) ...
Notes
... A topological space is irreducible if it cannot be written as a union of proper closed subsets. In particular, irreducible spaces are connected. The underlying topological space of a scheme is sober, i.e., every irreducible closed subspace has a generic point. Let A be a ring. Closed points of Spec( ...
... A topological space is irreducible if it cannot be written as a union of proper closed subsets. In particular, irreducible spaces are connected. The underlying topological space of a scheme is sober, i.e., every irreducible closed subspace has a generic point. Let A be a ring. Closed points of Spec( ...
Sec 5: Affine schemes
... Note that sections of the stalk bundle over any U ⇢ X form a ring (but not a local ring) by pointwise addition and multiplication: s1 + s2 and s1 s2 are given by (s1 + s2 )(x) = s1 (x) + s2 (x) and (s1 s2 )(x) = s1 (x)s2 (x). For every basic open set Xf , let (Xf , oX ) = Rf . Each element a = frm 2 ...
... Note that sections of the stalk bundle over any U ⇢ X form a ring (but not a local ring) by pointwise addition and multiplication: s1 + s2 and s1 s2 are given by (s1 + s2 )(x) = s1 (x) + s2 (x) and (s1 s2 )(x) = s1 (x)s2 (x). For every basic open set Xf , let (Xf , oX ) = Rf . Each element a = frm 2 ...
Algebraic Groups I. Homework 10 1. Let G be a smooth connected
... (i) Define a unique PGL2 -action on SL2 lifting conjugation. Prove a k-automorphism of G preserving the standard Borel k-subgroup and the diagonal k-torus is induced by the action of a diagonal k-point of PGL2 . (ii) Prove that the homomorphism PGL2 (k) → Autk (G) is an isomorphism. In particular, e ...
... (i) Define a unique PGL2 -action on SL2 lifting conjugation. Prove a k-automorphism of G preserving the standard Borel k-subgroup and the diagonal k-torus is induced by the action of a diagonal k-point of PGL2 . (ii) Prove that the homomorphism PGL2 (k) → Autk (G) is an isomorphism. In particular, e ...
here
... Now, suppose the former. Then again, by closure under addition, v1 +v2 −v1 ∈ W1 . Thus, v2 ∈ W1 . Since we chose v2 ∈ W2 , we have shown than W1 ⊇ W2 . Now, suppose that v1 + v2 ∈ W2 . Then by the same reasoning, we have that v1 + v2 − v2 ∈ W2 . Thus, v1 ∈ W2 . But v1 was chosen arbitrarily in w1 . ...
... Now, suppose the former. Then again, by closure under addition, v1 +v2 −v1 ∈ W1 . Thus, v2 ∈ W1 . Since we chose v2 ∈ W2 , we have shown than W1 ⊇ W2 . Now, suppose that v1 + v2 ∈ W2 . Then by the same reasoning, we have that v1 + v2 − v2 ∈ W2 . Thus, v1 ∈ W2 . But v1 was chosen arbitrarily in w1 . ...
LECTURE 2 Defintion. A subset W of a vector space V is a subspace if
... are called linear combinations. So a non-empty subset of V is a subspace if it is closed under linear combinations. Much of today’s class will focus on properties of subsets and subspaces detected by various conditions on linear combinations. Theorem. If W is a subspace of V , then W is a vector spa ...
... are called linear combinations. So a non-empty subset of V is a subspace if it is closed under linear combinations. Much of today’s class will focus on properties of subsets and subspaces detected by various conditions on linear combinations. Theorem. If W is a subspace of V , then W is a vector spa ...
Vector Spaces and Linear Maps
... often drawn as an arrow in the plane from the origin to (x1 , x2 ), or sometimes as a translation of this arrow. The sum of vectors can be seen as the result of joining them “tip to tail”. ...
... often drawn as an arrow in the plane from the origin to (x1 , x2 ), or sometimes as a translation of this arrow. The sum of vectors can be seen as the result of joining them “tip to tail”. ...
2.1. Functions on affine varieties. After having defined affine
... of K(X) of all functions that are regular (i. e. well-defined) at P, and for U ⊂ X an open subset we let OX (U) be the subring of K(X) of all functions that are regular at every P ∈ U. The ring of functions that are regular on all of X is precisely A(X). Given two ringed spaces (X, OX ), (Y, OY ) wi ...
... of K(X) of all functions that are regular (i. e. well-defined) at P, and for U ⊂ X an open subset we let OX (U) be the subring of K(X) of all functions that are regular at every P ∈ U. The ring of functions that are regular on all of X is precisely A(X). Given two ringed spaces (X, OX ), (Y, OY ) wi ...
Non-archimedean analytic geometry: first steps
... to view the space A1 as the affine line was the fact that it provided an answer to the question on the spectrum of a bounded linear operator I posed to myself at the very beginning. Namely, the spectrum of such an operator is a nonempty compact subset of A1 , and it coincides with the complement of ...
... to view the space A1 as the affine line was the fact that it provided an answer to the question on the spectrum of a bounded linear operator I posed to myself at the very beginning. Namely, the spectrum of such an operator is a nonempty compact subset of A1 , and it coincides with the complement of ...
18. Fibre products of schemes The main result of this section is
... Xi . Now construct a sheaf OX on X, using (18.3). This gives us a locally ringed space (X, OX ) and the remaining properties can be easily checked. ...
... Xi . Now construct a sheaf OX on X, using (18.3). This gives us a locally ringed space (X, OX ) and the remaining properties can be easily checked. ...
HYPERBOLIC GEOMETRY HANDOUT I MATH 180 In this
... Proof of (6). Möbius transformations are clearly bijections, since they have inverses that are also maps from Ĉ → Ĉ. We need to see that Möbius transformations map open sets to open sets. It suffices to show that open balls are mapped to open sets by affine maps and by inversion in the circle. T ...
... Proof of (6). Möbius transformations are clearly bijections, since they have inverses that are also maps from Ĉ → Ĉ. We need to see that Möbius transformations map open sets to open sets. It suffices to show that open balls are mapped to open sets by affine maps and by inversion in the circle. T ...
Solution 8 - D-MATH
... OX,p = {(f, U ) : p ∈ U ⊂ X nonempty open, f : U → C algebraic}/ ∼, where (f, U ) ∼ (g, V ) if there exists a nonempty open neighborhood W ⊂ U ∩V of p with f |W = g|W . Show that OX,p has a natural C-algebra structure. It is called the ring of germs of algebraic functions on X around p. If V ⊂ X is ...
... OX,p = {(f, U ) : p ∈ U ⊂ X nonempty open, f : U → C algebraic}/ ∼, where (f, U ) ∼ (g, V ) if there exists a nonempty open neighborhood W ⊂ U ∩V of p with f |W = g|W . Show that OX,p has a natural C-algebra structure. It is called the ring of germs of algebraic functions on X around p. If V ⊂ X is ...
PDF
... The resulting category is called the category of schemes over Y , and is sometimes denoted Sch/Y . Frequently, Y will be the spectrum of a ring (or especially a field) R, and in this case we will also call this the category of schemes over R (rather than schemes over Spec R). Observe that this resol ...
... The resulting category is called the category of schemes over Y , and is sometimes denoted Sch/Y . Frequently, Y will be the spectrum of a ring (or especially a field) R, and in this case we will also call this the category of schemes over R (rather than schemes over Spec R). Observe that this resol ...
16. Subspaces and Spanning Sets Subspaces
... some basic set of vectors? How do we change our point of view from vectors labeled one way to vectors labeled in another way? Let’s start at the top. ...
... some basic set of vectors? How do we change our point of view from vectors labeled one way to vectors labeled in another way? Let’s start at the top. ...
Affine Varieties
... OX (U ) := {φ ∈ C(X) | φ is regular at all points of U } for Zariski-open subsets U ⊆ X. Sheaf Overview: A sheaf of abelian groups on a topological space X is, first of all, a contravariant functor: F : {open subsets of X} → {abelian groups} from the category of open subsets of X to the category of ...
... OX (U ) := {φ ∈ C(X) | φ is regular at all points of U } for Zariski-open subsets U ⊆ X. Sheaf Overview: A sheaf of abelian groups on a topological space X is, first of all, a contravariant functor: F : {open subsets of X} → {abelian groups} from the category of open subsets of X to the category of ...
2.1
... • Sum of subsets S1, S2, …,Sk of V • If Si are all subspaces of V, then the above is a subspace. • Example: y=x+z subspace: • Row space of A: the span of row vectors of A. • Column space of A: the space of column vectors of A. ...
... • Sum of subsets S1, S2, …,Sk of V • If Si are all subspaces of V, then the above is a subspace. • Example: y=x+z subspace: • Row space of A: the span of row vectors of A. • Column space of A: the space of column vectors of A. ...
EXTERNAL DIRECT SUM AND INTERNAL DIRECT SUM OF
... Definition 1.1. The vector space V ⊕e W over F defined above is called the external direct sum of V and W. Let Z be a vector space over F and X and Y be vector subspaces of Z. Suppose that X and Y satisfy the following properties: (1) for each z ∈ Z, there exist x ∈ X and y ∈ Y such that z = x + y; ...
... Definition 1.1. The vector space V ⊕e W over F defined above is called the external direct sum of V and W. Let Z be a vector space over F and X and Y be vector subspaces of Z. Suppose that X and Y satisfy the following properties: (1) for each z ∈ Z, there exist x ∈ X and y ∈ Y such that z = x + y; ...
Chapter 2: Vector spaces
... • Sum of subsets S1, S2, …,Sk of V • If Si are all subspaces of V, then the above is a subspace. • Example: y=x+z subspace: • Row space of A: the span of row vectors of A. • Column space of A: the space of column vectors of A. ...
... • Sum of subsets S1, S2, …,Sk of V • If Si are all subspaces of V, then the above is a subspace. • Example: y=x+z subspace: • Row space of A: the span of row vectors of A. • Column space of A: the space of column vectors of A. ...
Regular differential forms
... 2 . It follows that there are no regular differential forms on P . Example Take the projective curve Y 2 Z = X 3 + XZ 2 . The projective plane P2 is covered by three affine pieces: A1 is the part with Z 6= 0 and coordinates (X, Y, 1), A2 is the part with Y 6= 0 and coordinates (U, 1, V ), A3 is the ...
... 2 . It follows that there are no regular differential forms on P . Example Take the projective curve Y 2 Z = X 3 + XZ 2 . The projective plane P2 is covered by three affine pieces: A1 is the part with Z 6= 0 and coordinates (X, Y, 1), A2 is the part with Y 6= 0 and coordinates (U, 1, V ), A3 is the ...
Homework - BetsyMcCall.net
... g. A subset H of a vector space V is a subspace of V if the following conditions are satisfied: i) the zero vector of V is in H, ii) u , v and u v are in H, and iii) c is a scalar and cu is in H. ...
... g. A subset H of a vector space V is a subspace of V if the following conditions are satisfied: i) the zero vector of V is in H, ii) u , v and u v are in H, and iii) c is a scalar and cu is in H. ...
Polynomials over finite fields
... Similarly, product of any two elements from U is also from U by distributivity. Let |U| = p, finite. Then U is isomorphic with Z/pZ with respect to. addition and multiplication. In this case p is a prime, otherwise F would have a zero divisor, so U= Fp. And Fp is also called the prime subfield of F. ...
... Similarly, product of any two elements from U is also from U by distributivity. Let |U| = p, finite. Then U is isomorphic with Z/pZ with respect to. addition and multiplication. In this case p is a prime, otherwise F would have a zero divisor, so U= Fp. And Fp is also called the prime subfield of F. ...
Algebraic Geometry
... open covering by affine schemes. There is a natural functor V 7! V from the category of algebraic spaces over k to the category of schemes of finite-type over k, which is an equivalence of categories. The algebraic varieties correspond to geometrically reduced schemes. To construct V from V , on ...
... open covering by affine schemes. There is a natural functor V 7! V from the category of algebraic spaces over k to the category of schemes of finite-type over k, which is an equivalence of categories. The algebraic varieties correspond to geometrically reduced schemes. To construct V from V , on ...
4. Morphisms
... g. If we assume that I is radical (which is the same as saying that R does not have any nilpotent elements except 0) then X = V (I) is an affine variety in An with coordinate ring A(X) ∼ = R. Note that this construction of X from R depends on the choice of generators of R, and so we can get differen ...
... g. If we assume that I is radical (which is the same as saying that R does not have any nilpotent elements except 0) then X = V (I) is an affine variety in An with coordinate ring A(X) ∼ = R. Note that this construction of X from R depends on the choice of generators of R, and so we can get differen ...
Affine space
In mathematics, an affine space is a geometric structure that generalizes certain properties of parallel lines in Euclidean space. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors between two points of the space. Thus it makes sense to subtract two points of the space, giving a vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a vector to a point of an affine space, resulting in a new point displaced from the starting point by that vector. The simplest example of an affine space is a linear subspace of a vector space that has been translated away from the origin. In finite dimensions, such an affine subspace corresponds to the solution set of an inhomogeneous linear system. The displacement vectors for that affine space live in the solution set of the corresponding homogeneous linear system, which is a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space.