PPT-Chapter 12 Vectors and the Geometry of Space

Section 121 ThreeDimensional Coordinate Systems Section 122 Vectors Section 123 The Dot Product Section 124 The Cross Product Section 125 Lines and Planes in Space Section 126 Cylinders and . . Anthony Lasenby. Astrophysics Group. Cavendish Laboratory. Cambridge, UK. a.n.lasenby@mrao.cam.ac.uk. www.mrao.cam.ac.uk/~clifford. Overview. Want to share two recent exciting developments. Recent progress in cosmology. Section 3.4. Bases for Subspaces. Spanning Sets. Let . W. be a subspace of . , and . a set of vectors in . W. (. i.e. . ). The set . S. is a . spanning set . for . W. (or . S. . spans. . W. ) if and only if every vector in . Real Vector Spaces. Subspaces. Linear Independence. Basis and Dimension. Row Space, Column Space, and Nullspace. Rank and Nullity. 2. 5-2 Subspaces. A . subset. . W. of a vector space . V. is called a . Fundamental system in linear algebra : system of linear equations . A. x . = . b. . nice case – . n. equations, . n. unknowns. matrix notation. row picture. column picture. linear combinations. For our matrix, can I solve . Daniel Svozil. based on excelent video lectures by Gilbert Strang, MIT. http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/VideoLectures/index.htm. Lectur. e. 5, Lecture 6. Transposes. How to write tra. John Hannah (Canterbury, NZ). Sepideh. Stewart (Oklahoma, US) . Mike Thomas (Auckland, NZ). Summary. Goals for a linear algebra course. Experiments and writing tasks. Examples. Student views. What do today’s students need?. & Subspaces. Kristi Schmit. Definitions. A subset W of vector space V is called a . subspace . of V . iff. The. . zero vector of V is in W.. W. is closed under vector addition, for each . u. . Similarity Search. Alex Andoni. (Columbia University). Find pairs of similar images. 2. how should we measure similarity?. Naïvely: about . comparisons. Can we do better?.  . Measuring similarity. ARM Research. 9. . Unification. Euclidean geometry. L9 . S. 2. Represent the Euclidean point . x. by null vectors. Distance is given by the inner product. Read off the Euclidean vector. D. epends on the concept of the origin. Vector Spaces. MATH . 264 Linear . Algebra. Introduction. There are two types of physical quantities:. Scalars = quantities that can be described by numerical value alone (Ex: temperature, length, speed). 4.1 Vectors in . R. n. 4.2 Vector Spaces. 4.3 Subspaces of Vector Spaces. 4.4 Spanning Sets and Linear Independence. 4.5 Basis and Dimension. 4.6 Rank of a Matrix and Systems of Linear Equations. 3D Coordinate System. Vectors in Space. Real life applications. Review from 10.1. Two vectors are equal if they have the same magnitude and direction (same length and are parallel). The zero vector, denoted by 0, is a vector with zero magnitude (undefined direction). Take your name card from the table. Place it on your desk with your name facing the . instructor. Fill out the information form that is on your . desk . Sit quietly. Honors Geometry. Dr. Alan L. . Breitler. Does space exist?. What is the relation between a theory and reality?. Views of Reality: a spectrum. The common person. There are definite events independent of observation. Our senses record these events. Theories can represent genuine causal patterns inherent in the events. Generally, the features we use to describe things, e.g. size, time…, are inherent in the events themselves. The world consists of collections of 'things'..