space向量空间.pptVIP

7 - * 7 - * If S={v1, v2,…, vk} is a set of vectors in a vector space V, then the span of S is the set of all linear combinations of the vectors in S, the span of a set: span (S) a spanning set of a vector space: If every vector in a given vector space can be written as a linear combination of vectors in a given set S, then S is called a spanning set of the vector space. 7 - * Notes: Notes: 7 - * Ex 5: (A spanning set for R3) Sol: 7 - * Lecture 7 Vector Space Last Time - Properties of Determinants - Introduction to Eigenvalues - Applications of Determinants - Vectors in Rn Elementary Linear Algebra R. Larsen et al. (5 Edition) TKUEE翁慶昌-NTUEE SCC_10_2007 7 - * Lecture 6: Eigenvalue and Vectors Today Vector Spaces and Applications Subspaces of Vector Spaces Spanning Sets and Linear Independence Basis and Dimension Rank of a Matrix and Systems of Linear Equations Reading Assignment: Secs. 4.2-4.6 of Textbook Homework #3 due Next Time Rank of a Matrix and Systems of Linear Equations (Cont.) Coordinates and Change of Basis Applications of Vector Spaces Length and Dot Product in Rn Inner Product Spaces Reading Assignment: Secs 4.7- 5.2 7 - * Lecture 6: Elementary Matrices Determinants Today Vector Spaces Subspaces of Vector Spaces Spanning Sets and Linear Independence Basis and Dimension Rank of a Matrix and Systems of Linear Equations 7 - * What Did You Actually Learn about Determinant? 7 - * 4.2 Vector Spaces Vector spaces: Let V be a set on which two operations (vector addition and scalar multiplication) are defined. If the following axioms are satisfied for every u, v, and w in V and every scalar (real number) c and d, then V is called a vector space. Addition: (1) u+v is in V (2) u+v=v+u (3) u+(v+w)=(u+v)+w (4) V has a zero vector 0 such that for every u in V, u+0=u (5) For every u in V, there is a vector in V denoted by –u such that u+(–u)=0 7 - * Scalar multiplication: (6) is in V.