1.1 Limits and Continuity. Precise definition of a limit and limit laws
... Definition 2.2 The characteristic polynomial of a linear operator T is the polynomial PA (λ) = det(A−λI), where A is the matrix of T with respect to any basis. The statement below follows from the fact that similar matrices represent the same linear operator. Corollary 2.4 Similar matrices have the ...
... Definition 2.2 The characteristic polynomial of a linear operator T is the polynomial PA (λ) = det(A−λI), where A is the matrix of T with respect to any basis. The statement below follows from the fact that similar matrices represent the same linear operator. Corollary 2.4 Similar matrices have the ...
TOPOLOGY TAKE-HOME 1. The Discrete Topology Let Y = {0,1
... Int(A ∪ B) ⊆ (IntA) ∪ (IntB). Proof. Let x ∈ Int(A ∪ B). Then, by Lemma 3.2, there is a neighborhood U of x such that U ⊆ A ∪ B. Now, either x ∈ A or x ∈ B. Suppose, without loss of generality, that x ∈ A. Then x ∈ A ∩ U ⊆ A, so x ∈ IntA. Hence, x ∈ (IntA)∪(IntB). Since our choice of x was arbitrary ...
... Int(A ∪ B) ⊆ (IntA) ∪ (IntB). Proof. Let x ∈ Int(A ∪ B). Then, by Lemma 3.2, there is a neighborhood U of x such that U ⊆ A ∪ B. Now, either x ∈ A or x ∈ B. Suppose, without loss of generality, that x ∈ A. Then x ∈ A ∩ U ⊆ A, so x ∈ IntA. Hence, x ∈ (IntA)∪(IntB). Since our choice of x was arbitrary ...
aa1
... injective. (You can use the property of a compact Hausdorff space X that a C-valued continuous function on a closed subset C of X extends to a C-valued continuous function on X.) 9. Prove that X is connected if and only if there is no f ∈ A such that f 2 = f , f 6= 0, f 6= 1. 10. Assume X is a finit ...
... injective. (You can use the property of a compact Hausdorff space X that a C-valued continuous function on a closed subset C of X extends to a C-valued continuous function on X.) 9. Prove that X is connected if and only if there is no f ∈ A such that f 2 = f , f 6= 0, f 6= 1. 10. Assume X is a finit ...
Algebraic Geometry 3-Homework 11 1. a. Let O be a noetherian
... Show that O is a normal ring. Show that O is a discrete valuation ring (DVR) if O has Krull dimension one. b. Let X be an integral k-scheme. X is called locally factorial if each local ring OX,x is a UFD (for example, a smooth k-scheme is locally factorial). Show that for a locally factorial k-schem ...
... Show that O is a normal ring. Show that O is a discrete valuation ring (DVR) if O has Krull dimension one. b. Let X be an integral k-scheme. X is called locally factorial if each local ring OX,x is a UFD (for example, a smooth k-scheme is locally factorial). Show that for a locally factorial k-schem ...
CW Complexes and the Projective Space
... Remark: Depending on the nature of the functions considered in the atlas (smooth, analytic, polinomial, etc.) we can regard CP n as a smooth manifold, complex manifold, ...
... Remark: Depending on the nature of the functions considered in the atlas (smooth, analytic, polinomial, etc.) we can regard CP n as a smooth manifold, complex manifold, ...
