矩陣分析 引文 第一講

1、1Lecture 1:Lecture 1: Vector Spaces and Vector NormsVector Spaces and Vector Norms MATRIX ANALYSIS HITSZTIME: Autumn 2011INSTRUCTOR: You-Hua FanReading assignment on the textbook Section 0.1 Section 5.1, 5.2, 5.4.1-5.4.7 2345678910(f)(ba111212k1314151617181V19V20One of the most remarkable features o

2、f vector spaces is the notion of dimension.We need one simple result that makes this happen, the basis theorem.21222324Norms are a way of putting a measure of distance on vector spaces. The purpose is for the refined analysis of vector spaces from the viewpoint of many applications. It is also to al

3、l the comparison of various vectors on the basis of their length. 25Norms derived from inner product:1.2.26If , we can proof that it is a vector norm.Firstly, we will proof the 27Then, we proof the With the triangle inequality, we can easily proof thatis a vector norm. we say it is derived from the

4、inner product. 28In fact, (i) is a norm derived from the standard inner product:2930=313233 34353637Basic concepts: vector space, subspace, span, linear combination, linearly independent, linear dependent, basis, dimension, vector norm, inner product.Important principles: *A span is a subspace. *Zer

5、o vector is l.d. to all vectors; *Subset of a l.i. set is l.i.; *L.i. vectors can be added to form a basis; *Every basis has the same number of vectors; *Each vector has a unique basis-representation; *Every inner product has the Cauchy-Schwarz inequality. * Every inner product can be used to define a vector norm; * Every vector norm is a continuous function; *All vector norms