Lecture 1 : the geometry of linear equati 4000 ons

Lecture 1 : the geometry of linear equations

教材：Introduction to Linear Algebra, Gilbert Strang, 4th

网站：ocw.mit.edu \ web.mit.edu/18.06 \ math.mit.edu/linearalgebra

Preface

The crucial operation in linear algebra is taking linear combinations of vectors.

Chapter 1 Introduction to Vectors

1.1 Vectors and Linear Combinations

The heart of linear algebra is in two operations – vector addition and scalar multiplication, combining them is linear combinations.

维度问题：m个向量可以构成一个m维空间（一般来说），如dot(0)、line(1)、plane(2)、space(3)等。

矢量加法遵循三角形定则。

Linear combinations 满足交换律及结合律。

1.2 Lengths and Dot Products

DEFINITION The dot product or inner product of v=(v1,v2) and w=(w1,w2)

is the number v⋅w :

v⋅w=v1w1+v2w2.

点乘满足交换律。

In mathematics, zero is always a special number. For dot products, it means that these two vectors are perpendicular. The angle between them is 90°.

Main point To compute v⋅w , multiply each vi times wi Then add ∑viwi.

DEFINITION The length ∥v∥ of a vector v is the square root of v⋅v:

Length = norm(v) length = ∥v∥ = v⋅v−−−−√.