for twoside printing - Institute for Statistics and Mathematics
... number is even. When reading the definition of even number we find the terms divisible and natural numbers. These terms must already be well-defined: We say that a natural number n is divisible by a natural number k if there exists a natural number m such that n = k · m. What are natural numbers? Th ...
... number is even. When reading the definition of even number we find the terms divisible and natural numbers. These terms must already be well-defined: We say that a natural number n is divisible by a natural number k if there exists a natural number m such that n = k · m. What are natural numbers? Th ...
Notes 4: The exponential map.
... Notes 4: The exponential map. Version 0.00 — with misprints, Recap of vector fields. Recall that a smooth vector field , or just a vector field for short, on G is a section of the tangent bundle; in other words, it is a smooth map v : G → T G such that vπ = idG . The map v picks out a tangent vector ...
... Notes 4: The exponential map. Version 0.00 — with misprints, Recap of vector fields. Recall that a smooth vector field , or just a vector field for short, on G is a section of the tangent bundle; in other words, it is a smooth map v : G → T G such that vπ = idG . The map v picks out a tangent vector ...
Linear Spaces
... Lines are among the fundamental objects in an Euclidean space. a line is determined by a point x0 and a direction v. To determine any point on the line we add scalar multiples of v to x0 , we obtain the parametric representation of the line x(t) = x0 + tv. ...
... Lines are among the fundamental objects in an Euclidean space. a line is determined by a point x0 and a direction v. To determine any point on the line we add scalar multiples of v to x0 , we obtain the parametric representation of the line x(t) = x0 + tv. ...
CLASS NOTES ON LINEAR ALGEBRA 1. Matrices Suppose that F is
... Span(S) = {c1 v1 + · · · + cn vn | n ∈ Z+ , v1 , . . . , vn ∈ S and c1 , . . . , cn ∈ F }. This definition is valid even when S is an infinite set. We define Span{∅} = {~0}. Lemma 2.2. Suppose that S is a subset of V . Then Span(S) is a subspace of V . Definition 2.3. Suppose that S is a subset of V ...
... Span(S) = {c1 v1 + · · · + cn vn | n ∈ Z+ , v1 , . . . , vn ∈ S and c1 , . . . , cn ∈ F }. This definition is valid even when S is an infinite set. We define Span{∅} = {~0}. Lemma 2.2. Suppose that S is a subset of V . Then Span(S) is a subspace of V . Definition 2.3. Suppose that S is a subset of V ...
