线性代数 英文讲义

Chapter 4 Linear Transformations In this chapter, we introduce the general concept of linear transformation from a vector space into a vector space. But, we mainly focus on linear transformations from n R to m R.

§1 Definition and Examples

New words and phrases

Mapping 映射

Linear transformation 线性变换

Linear operator 线性算子

Dilation 扩张

Contraction 收缩

Projection 投影

Reflection 反射

Counterclockwise direction 反时针方向

Clockwise direction 顺时针方向

Image 像

Kernel 核

1.1 Definition

★Definition A mapping(映射) L: V W is a rule that produces a correspondence between two sets of elements such that to each element in the first set there corresponds one and only one element in the second set.

★Definition A mapping L from a vector space V into a vector space W is said to be a linear transformation（线性变换）if

(1) 11221122(v v )(v )(v )L L L αααα+=+

for all 12v ,v V ∈ and for all scalars 1α and 2α. (1) is equivalent to

(2) 1212(v v )(v )(v )L L L +=+ for any 12v ,v V ∈ and

(3) (v)(v)L L αα= for any v V ∈ and scalar α.

Notation: A mapping L from a vector space V into a vector space W is denoted

L: V →W

When W and V are the same vector space, we will refer to a linear transformation L: V →V as a linear operator on V . Thus a linear operator is a linear transformation that maps a vector space V into itself.

1.2 Linear Operators on 2R

1. Dilations(扩张) and Contractions Let L be the operator defined by

L(x)=k x

then this is a linear operator. If k is a positive scalar, then the linear operator can be thought of as a stretching or shrinking by a factor of k.

120(x)x 0x k L A x k ????== ? ?????

2. Projection （投影）onto the coordinate axes.

L(x)=11e x 1210(x)x 00x L A x ??

??== ?

????? L(x)=22e x 1200(x)x 01x L A x ??

??== ?

?????

3. Reflections (反射) about an axis Let L be the operator defined by

L(x)=12(,)T x x -, then it is a linear operator. The operator L has the

effect of reflecting vectors about the x-axis. 1210(x)x 01x L A x ????== ? ?-????

Reflecting about the y-axis

L(x)=12(,)T

x x -, 1210(x)x 01x L A x -??

??== ? ?????

4. Rotations

L(x)=21(,)T x x -, L has the effect of rotating each vector by 90 degrees in the counterclockwise direction （逆时针方向）.

1201(x)x 10x L A x -??

??== ? ?????

1.3 Linear Transformations from n R to m R

If A is an mxn matrix, then we can define a linear transformation A L from n R to m R by

()A L X AX =

It is easy to verify that the mapping above is linear. In the next section, we will see that any linear transformation from n R to m R must be of this form.

1.4 The Image and Kernel

★Definition Let L: n R →m R is a linear transformation. The kernel （核）of L denoted ker(L), is defined by ker(L)={}v |(v)0W V L ∈=

★Definition Let L: n R →m R is a linear transformation and let S be a subspace of V . The image （像）of S, denoted L(S), is defined by L(S)= {}w |w (v) for some v m n R L R ∈=∈

The image of the entire vector space, L(V), is called the range （值域）of L.

Theorem 4.1.1 If L: n R →m R is a linear transformation and S is a subspace of n R , then (i) ker(L) is a subspace of n R . (ii) L(S) is a subspace of m R .

Assignment for section 1, chapter 4

Hand in: 3, 4, 17, 20,

Not required : 8, 10, 11, 15, 16, 19, 25

§2 Matrix Representations of Linear Transformations

New words and phrases

Matrix representation 矩阵表示 Formal multiplication 形式乘法 Similarity 相似性

2.1 Matrix Representation of Linear Transformations

In section 1 of this chapter, the examples of linear transformations can be represented by matrices. In general, a linear transformation can be represented by a matrix.

If we use the basis E=[12u ,u ,,u n ] for U and the basis F=[12v ,v ,,v m ] for V , and L: U → V . If u is a vector in U, then

1122u u u u n n x x x =+++ (in U) |→ 1122L(u)v v v m m y y y =+++ (in V)

The linear transformation L is determined by the change of the coordinate vectors:

12n x x x ?? ? ? ? ??? →12m y y y ?? ? ? ? ???

Assume that

1122(u )v v v j j j mj m L a a a =+++ , j=1, 2, …, n

Formally,

12[(u ),(u ),,(u )]n L L L =12[v ,v ,,v ]m 1112121

2221

2

n n m m mn a a a a a a a a a ??

?

?

? ?

??

Write 12[(u ),(u ),,(u )]n L L L as linear combinations of 12[v ,v ,,v ]m , then consequently, A is obtained.

Then L(u) = 1122(u u u )n n L x x x +++ =12[v ,v ,,v ]m 12m y y y ?? ? ? ? ???

(formal multiplication)

1122L(u )L(u )L(u )n n x x x =+++ =n

j=1(u )j j x L ∑

1212[L(u ),L(u ),,L(u )]n n x x

x ?? ? ?= ? ???

(formal multiplication)

==1

1

(v )n m

j ij i j i x a =∑∑

=i=11

()v m n ij j i j a x =∑∑=12[v ,v ,,v ]m 1112121

2221

2

n n m m mn a a a a

a a a a a ??

?

? ?

?

?? 12n x x x ?? ? ? ? ???

Hence

12m y y y ?? ? ? ? ??? =1112121

2221

2

n n m m mn a a a a

a a a a a ??

?

? ?

?

??

12n x x x ?? ? ? ? ???

Thus, y=A x is the coordinate vector of L (u) with respect to

F =[12v ,v ,,v m ]. y=A x is called the matrix representation of the linear transformation. A is called the matrix representing L relative to the

bases E and F. A is determined by the following equations.

12[(u ),(u ),,(u )]n L L L =12[v ,v ,,v ]m 1112121

2221

2

n n m m mn a a a a

a a a a a ??

?

?

?

?

??

We have established the following theorem.

Theorem 4.2.2 If E=[12u ,u ,,u n ] is an ordered basis for U and F=[12v ,v ,,v m ] is an ordered basis for V , then corresponding to each linear transformation L:U →V there is an mxn matrix A such that [()][]F E L u A u = for each u in U.

A is the matrix representing L relative to the ordered bases E and F. In fact, a [(u )]j j F L =.

2.2 Matrix Representation of L: n R →m R

If U=n R , V=m R , then we have the following theorem.

Theorem 4.2.1 If L is a linear transformation mapping n R into m R , there is an mxn matrix A such that

L (x)=A x

for each x n R ∈. In fact, the jth column vector of A is given by

12((e ),(e ),,(e ))n A L L L =

Proof If we choose standard basis 12[e ,e ,,e ]n for n R and the

standard basis 12[e ,e ,,e ]m for m R ,

L(x)= 1122(e e e )n n L x x x +++ =12(e ,e ,,e )m 12m y y y ?? ? ? ? ??? =12m y y

y ?? ? ? ? ???

112

2L (e )L (e )L (e )n

n

x x x =+

++ =n

j=1

(e )j j x L ∑

1212(L(e ),L(e ),,L(e ))n n x x x ??

? ?= ? ???

And let A=()ij a =()12a ,a ,,a n 12(L(e ),L(e ),,L(e ))n = If 1122x e e e n n x x x =+++ , then L(x)=Ax.

A is referred to as the standard matrix representation (标准矩阵表示) of L.( A representation with respect to the standard basis.)

Example 1 (example 1 on page 186) Determine the standard matrix representation of L.

Define the linear transformation L:3R 2R by

1223(x)(,)T L x x x x =++ for each 123x (,,)T x x x = in 3R , find the linear standard representation of L.

Solution: Find 123L(e ),L(e ),L(e ). Then 123110(L(e ),L(e ),L(e ))011A ??

== ???

Example 2 rotation by an angle θ

Let L be the linear transformation operator on 2R that rotates each vector by an angle θ in the counterclockwise direction. We can see that

1e is mapped to (cos ,sin )T θθ, and 2e is mapped to (sin ,cos )T θθ-.

1(e )(cos ,sin )T L θθ=, 2(e )(sin ,cos )T L θθ=-

The matrix A representing the transformation will be

12cos sin (L(e ),L(e ))sin cos A θ

θθ

θ-??

==

???

To find the matrix representation A for a linear transformation L n R

→

m R w.r.t. the bases E=[12u ,u ,,u n ] and F=[12b ,b ,,b m ], we

must represent each vector 1122(u )b b b j j j mj m L a a a =+++ . The following theorem shows that determining this representation is equivalent to solving the linear system Bx=(u )j L , where (u )j L is regarded as a column vector in m R .

Theorem 4.2.3 Let E =[12u ,u ,,u n ] and F =[12b ,b ,,b m ] be ordered bases for n R and m R , respectively. If L : n R → m R is a linear transformation and A is the matrix representing L with respect to E and F , then

112((u ),(u ),,(u ))n A B L L L -=

where B =(12b ,b ,,b m ).

Proof L(u) = 1122(u u u )n n L x x x +++ =12(b ,b ,,b )m 12m y y y ??

? ? ? ???

1122L(u )L(u )L(u )n n x x x =+++ =n

j=1

(u )j j x L ∑

1212(L(u ),L(u ),,L(u ))n n x x

x ?? ? ?= ? ???

12(b ,b ,,b )m 12m y y y ?? ?

? ?

?

??

1212(L(u ),L(u ),,L(u ))n

n x x x ??

? ?= ? ???

The matrix B is nonsingular since its column vectors form a basis for

m R . Hence, 112((u ),(u ),,(u ))n A B L L L -=

12((u ),(u ),,(u ))n L L L is the matrix representing L relative to the bases [12u ,u ,,u n ] and [12e ,e ,,e m ]. B is the transition matrix corresponding to the change of basis from [12b ,b ,,b m ] to [12e ,e ,,e m ].

Corollary 4.2.4 If A is the matrix representing the linear transformation L:

n R m R with respect to the bases

12m [b ,b ,,b ]

1B -

12m [e ,e ,,e ]

12n [u ,u ,,u ]

A

12((),(),,())n L u L u L u

B

E=[12u ,u ,,u n ] and F=[12b ,b ,,b m ]

then the reduced row echelon form of

1212(b ,b ,,b |(u ),L(u ),,L(u ))m n L

is (I |A )

Proof 1212(b ,b ,,b |(u ),L(u ),,L(u ))m n L =(B|BA), which is row equivalent to (I|A).

Examples Finding the matrix representing L Example 3 on page 188

Let L be a linear transformation mapping 3R into 2R defined by 1123L (x )b ()b x x x =++. Find the matrix A representing L with respect to the ordered bases 123[e ,e ,e ] and 12[b ,b ], where 11b 1??= ???, 21b 1-??

= ???

Solution:

Method 1. Represent 123[e ,e ,e ] in terms of 12[b ,b ] Method 2. 112123(b ,b )((e ),(e ),(e ))A L L L -=

1

111111/21/2111100111111/21/2111011A ------??????????=== ? ? ??? ?-??????????

Method 3 Applying row operations. 12123(b ,b |(e ),(e ),(e ))L L L Example 4 on page 188

Let L be a linear transformation mapping 2R into itself defined by 1212L (b b )()b 2b αβαββ+=++, where

12[b ,b ] is the ordered basis

defined in example 3. Find the matrix A representing L with respect to

12[b ,b ].

Solution: Use three methods as in example 3. Example 6 on page 190

Determine the matrix representation of L with respect to the given bases. Let L: 2R 3R be the linear transformation defined by

21212

L (x )(,,)T

x x x x x =+- Find the matrix representation of L with respect to the ordered bases

12[u ,u ] and 123[b ,b ,b ], where

12u (1,2),u (3,1)T T ==

123b (1,0,0),b (1,1,0),b (1,1,1)T T T ===

Assignment for section 2, chapter 4

Hand in: 2, 6, 8, 16, 20 Not required: 9—15, 17, 19

§3 Similarity

Let L be a linear operator on V , E=[12v ,v ,,v n ] be an ordered basis for V , A is the matrix representing L with respect to the basis E. 1122u v v v

n n x x x =+++ , 1122L(u)v v v n n y y y =+++ 1211221212(v )v v v [v ,v ,,v ][v ,v ,,v ]a j j j j j nj n n n j nj a a L a a a a ??

? ?=+++== ? ? ???

y=Ax

F=[12w ,w ,,w n ]

1122u w w w n n c c c =+++ , 1122L(u)w w w n n d d d =+++

1211221212(w )w w w [w ,w ,,w ][w ,w ,,w ]b j j

j j j nj n n n j nj b b L b b b b ?? ? ?=+++== ? ? ???

d=Bc

Let the transition matrix corresponding the change of basis from F=[12w ,w ,,w n ] to [12v ,v ,,v n ]

Then x=Sc, y=Sd, 11y x S BS --= 1y x SBS -= or 1A SBS -=

Hence, we have established the following theorem.

Theorem 4.3.1 Let E=[12v ,v ,,v n ] and F=[12w ,w ,,w n ] be two ordered bases for a vector space V , and let L be a linear operator on

n R . Let S be the transition matrix representing the change from F to E. If

A is the matrix representing L with respect to E, and

B is the matrix representing L with respect to F, then 1B S AS -=.

★Definition Let A and B be nxn matrices. B is said to be similar to A if there is a nonsingular matrix S such that 1B S AS -=.

Example 2 (on page 204)

Example Let L be the linear operator on 3R defined by L(x)=Ax, where

V V

V V

Basis E=12n [v ,v ,,v ] S

u →L(u)

u →L(u)

Ax=y

Bc=d

S -1

Coordinate vector of L(u): d

Basis F

Coordinate vector of L(u) :y Basis E

Coordinate vector of u: x

Coordinate vector of u: c Basis E=12n [w ,w ,,w ] x=Sc y=Sd

220112112?? ? ? ???

. Thus the matrix A represents L with respect to the standard basis for 3R . Find the matrix representing L with respect to the basis [123y ,y ,y ], where 1y (1,1,0)T =-, 2y (2,1,1)T =-, 3y (1,1,1)T =. Solution

D=000010004??

?

? ???

is the matrix representing L w.r.t the basis [123y ,y ,y ],. Or, we can find D using 1D Y AY -= 11x ()x=()x n n n A YDY YD Y --=

Using this example to show that it is desirable to find as simple as a representation as possible for a linear operator. In particular, if the operator can be represented by a diagonal matrix, this is usually preferred representation. It makes the computation of Dx and x n D easier.

Assignment for section 3, chapter 4

Hand in: 2, 3, 4, 8, 10, 15 Not required 5, 6

线性代数英文单词

线性代数英语词汇大集合 ========================================================================= A adjont(adjugate) of matrix A A 的伴随矩阵 augmented matrix A 的增广矩阵 B block diagonal matrix 块对角矩阵 block matrix 块矩阵 basic solution set 基础解系 C Cauchy-Schwarz inequality 柯西 - 许瓦兹不等式 characteristic equation 特征方程 characteristic polynomial 特征多项式 coffcient matrix 系数矩阵 cofactor 代数余子式 cofactor expansion 代数余子式展开 column vector 列向量 commuting matrices 交换矩阵 consistent linear system 相容线性方程组 Cramer's rule 克莱姆法则 Cross- product term 交叉项 D Determinant 行列式 Diagonal entries 对角元素 Diagonal matrix 对角矩阵 Dimension of a vector space V 向量空间 V 的维数 E echelon matrix 梯形矩阵 eigenspace 特征空间 eigenvalue 特征值 eigenvector 特征向量

eigenvector basis 特征向量的基 elementary matrix 初等矩阵 elementary row operations 行初等变换 F full rank 满秩 fundermental set of solution 基础解系 G grneral solution 通解 Gram-Schmidt process 施密特正交化过程 H homogeneous linear equations 齐次线性方程组I identity matrix 单位矩阵 inconsistent linear system 不相容线性方程组indefinite matrix 不定矩阵 indefinit quatratic form 不定二次型 infinite-dimensional space 无限维空间 inner product 内积 inverse of matrix A 逆矩阵 J K L linear combination 线性组合 linearly dependent 线性相关 linearly independent 线性无关 linear transformation 线性变换 lower triangular matrix 下三角形矩阵 M main diagonal of matrix A 矩阵的主对角matrix 矩阵

考研数学线性代数讲义

1.题设条件与代数余子式Aij或A*有关，则立即联想到用行列式按 行（列）展开定理以及AA*=A*A=|A|E. 2.若涉及到A.B是否可交换，即AB=BA，则立即联想到用逆矩阵的定 义去分析。 3.若题设n阶方阵A满足f（A）=0，要证aA+bE可逆，则先分解出 因子aA+bE再说。 4.若要证明一组向量a1，a2，…，as线性无关，先考虑用定义再说。 5.若已知AB=0，则将B的每列作为Ax=0的解来处理再说。 6.若由题设条件要求确定参数的取值，联想到是否有某行列式为零再说。 7.若已知A的特征向量ζ0，则先用定义Aζ0=λ0ζ0处理一下再说。 8.若要证明抽象n阶实对称矩阵A为正定矩阵，则用定义处理一下再说。 2010考研基础班线性代数 主讲：尤承业 第一讲基本概念 线性代数的主要的基本内容：线性方程组矩阵向量行列式等一．线性方程组的基本概念 线性方程组的一般形式为: 其中未知数的个数n和方程式的个数m不必相等. 线性方程组的解是一个n个数 C,2C, …, n C构成,它满足:当每个方程中 1 的未知数1x都用1C替代时都成为等式. 对线性方程组讨论的主要问题两个:

(1)判断解的情况. 线性方程组的解的情况有三种:无解,唯一解,无穷多解. 如果两条直线是相交的则有一个解；如果两条直线是重合的则有无穷多个解；如果两条直线平行且不重合则无解。 (2)求解,特别是在有无穷多解时求通解. 齐次线性方程组: 021====n b b b 的线性方程组.0,0,…,0 总是齐次线性方程组的解,称为零解. 因此齐次线性方程组解的情况只有两种:唯一解(即只要零解)和无穷多解(即有非零解). 二.矩阵和向量 1.基本概念 矩阵和向量都是描写事物形态的数量形式的发展. 矩阵由数排列成的矩形表格, 两边界以圆括号或方括号, m 行n 列的表格称为m ?n 矩阵. 这些数称为他的元素,位于第i 行j 列的元素称为(i,j)位元素. 5401 23-是一个2?3矩阵. 对于上面的线性方程组,称矩阵 mn m m n n a a a a a a a a a A 212222111211=和m mn m m n n b b b a a a a a a a a a A 21212222111211)(=β

数学英文词汇大全-微积分,线性代数,概率统计

微积分 第一章函数与极限 Chapter1 Function and Limit 集合set 元素element 子集subset 空集empty set 并集union 交集intersection 差集difference of set 基本集basic set 补集complement set 直积direct product 笛卡儿积Cartesian product 开区间open interval 闭区间closed interval 半开区间half open interval 有限区间finite interval 区间的长度length of an interval 无限区间infinite interval 领域neighborhood 领域的中心centre of a neighborhood 领域的半径radius of a neighborhood 左领域left neighborhood 右领域right neighborhood 映射mapping X到Y的映射mapping of X ontoY 满射surjection 单射injection 一一映射one-to-one mapping 双射bijection 算子operator 变化transformation 函数function 逆映射inverse mapping 复合映射composite mapping 自变量independent variable 因变量dependent variable 定义域domain 函数值value of function 函数关系function relation 值域range 自然定义域natural domain

线性代数introduction to vectors

An introduction to vectors Definition of a vector A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction. The direction of the vector is from its tail to its head. Two vectors are the same if they have the same magnitude and direction. This means that if we take a vector and translate it to a new position (without rotating it), then the vector we obtain at the end of this process is the same vector we had in the beginning. Two examples of vectors are those that represent force and velocity. Both force and velocity are in a particular direction. The magnitude of the vector would indicate the strength of the force or the speed associated with the velocity. We denote vectors using boldface as in a or b. Especially when writing by hand where one cannot easily write in boldface, people will sometimes denote vectors using arrows as in a or b, or they use other markings. We won't need to use arrows here. We denote the magnitude of the vector a by ∥a∥. When we want to refer to a number and stress that it is not a vector, we can call the number a scalar. We will denote scalars with italics, as in a or b . You can explore the concept of the magnitude and direction of a vector using the below applet. Note that moving the vector around doesn't change the vector, as the position of the vector doesn't affect the magnitude or the direction. But if you stretch or turn the vector by moving just its head or its tail, the magnitude or direction will change. (This applet also shows the coordinates of the vector, which you can read about in another page.) The magnitude and direction of a vector. The blue arrow represents a vector a . The two defining properties of a vector, magnitude and direction, are illustrated by a red bar and a green arrow, respectively. The length of the red bar is the magnitude ∥a∥ of the vector a. The green arrow always has length one, but its direction is the direction of the vector a.

线性代数 英文讲义

Chapter 1 Matrices and Systems of Equations Linear systems arise in applications to such areas as engineering, physics, electronics, business, economics, sociology（社会学）, ecology （生态学）, demography(人口统计学), and genetics（遗传学）, etc. §1. Systems of Linear Equations New words and phrases in this section: Linear equation 线性方程 Linear system，System of linear equations 线性方程组 Unknown 未知量 Consistent 相容的 Consistence 相容性 Inconsistent不相容的 Inconsistence 不相容性 Solution 解 Solution set 解集 Equivalent 等价的 Equivalence 等价性 Equivalent system 等价方程组 Strict triangular system 严格上三角方程组 Strict triangular form 严格上三角形式 Back Substitution 回代法 Matrix 矩阵 Coefficient matrix 系数矩阵 Augmented matrix 增广矩阵 Pivot element 主元 Pivotal row 主行 Echelon form 阶梯形 1.1 Definitions A linear equation (线性方程) in n unknowns（未知量）is 1122... n n a x a x a x b +++=

线性代数讲义

线性代数讲义 线性代数攻略 线性代数由两部分组成: 第一部分:用矩阵解方程组(判断解的存在性,用有限个解表示所有的解)第二部分:用方程组解矩阵(求特征值,特征向量,对角化,化简实二次型)主观题对策 1. 计算题精解 计算题较之选择题与填空题难度几乎没有增加,但计算量大大增加,故出错的机会大幅增长,因此应力求用简便方法解决问题. 一.行列式的计算: 单纯计算行列式的题目大概永远不会出现.所以需要结合其它的知识点. l 核心内容 范德蒙行列式/余子式/代数余子式/Cramer法则: l 典型方法 降阶法(利用Gauss消元法化为三角矩阵:常常是将所有的行或列加到一起)/特征值法(矩阵的行列式等于其特征值之积)/行列式的其它性质(转置矩阵/逆矩阵/伴随矩阵/矩阵之积) 例1 计算下述三个n阶矩阵的行列式： . 解先算|B|=xn;再算|A|： 故|C|= |A|(-1)(1+?+n)+[(n+1)+…+(2n)] |B-1| =(-1)(1+2n)n(n+x)/x. 例2(2004-4) 设矩阵 ,矩阵B满足ABA*=2BA*+E,则|B|=[ ]. 分析化简可得(A-2E)BA*=E；于是|A-2E||B||A*|=1. 又|A*|=9,|A-2E|=1,所以|B|=1/9. (切忌算B=(A-2E)-1(A*)-1.) 例3 设4×4矩阵A=(x,a,b,g), B=(h,b,g,a). 若|A|=1, |B|=2,则行列式|A+B|=[ ].

正解：|A+B|=|x+h, a+b, b+g, g+a|=|x+h, 2(a+b+g), b+g, g+a|=2|x+h, a+b+g, b+g, g+a| =2|x+h, a, b+g, g+a|=2|x+h, a, b+g, g|=2|x+h, a, b, g|=2(|x, a, b, g|+|h, a, b, g|)=2(|A|+|B|)=6. 巧解:正解令人羡慕,但可能想不起来.于是令A=E,则.但|B|=2,所以取最简单的 .于是 ,故|A+B|=6. 例4 若四阶方阵A的特征值分别为-1,1,2,3,则行列式|A-1+2A*|=[ ]. 解此题考查对特征值的理解.特征值的性质中最重要(也是最简单的)的有两条，即所有特征值的和等于矩阵的迹(=对角线元素之和),而所有特征值的积等于矩阵的行列式.因此|A|= -6!剩余的就是简单的变形了: A-1+2A* = A-1 (E+2A A*) = A-1 (E+2|A|E)=-11A-1. 故|A-1+2A*|=|-11A-1|=(-11)4|A-1|=-114/6. 本题有巧解,你想到了吗?对!就让A是那个满足条件的最简单的矩阵! 例2(上海交大2002) 计算行列式 其中,. 本题只要对特征多项式有一定认识,则易如反掌.所求行列式对应的矩阵A=xE+B, 其中B=(aibj)的任意两行均成比例,故其秩为1(最重要的矩阵类型之一)或0,但由题中所给条件,B10,于是,B至少有n-1个特征值为0,另有一特征值等于trB= a1b1+ a2b2+…+ anbn10. 从而,A有n-1个特征值x,另有一个特征值x+trB.OK 例3(2001) 设A为三阶矩阵,X为三维向量,X,AX, A2X线性无关,A3X=4AX-3A2X.试计算行列式|2A2+3E|. 很多人觉得此题无从下手,实在冤枉了出题人.由A3X=2AX-3A2X可知, A(A2+3A-4E)X=0.由此知, |A|=0:否则,A可逆,X,AX, A2X将线性相关,矛盾!从而(A2+3A-4E)X=0:故X是齐次线性方程组(A2+3A-4E)Y=0的非零解.于是|A2+3A-4E|=0.故A的三个特征值为0,1,-4.于是2A2+3E的三个特征值为3,5,35.所以, |2A2+3E|=3′5′35=525. 例4(1995) 设n阶矩阵A满足AA￠=I,|A|<0,求|A+I|. 解首先, 1=|AA￠|=|A|2,所以|A|=-1. 其次, |A+I|=|A+AA￠|=|A||I+A￠|=|A||I+A|=-|I+A|, 故|A+I|=0. (涉及的知识点: |A|=|A￠|, (A+B)￠=A￠+B￠.) 例5(1999)设A是m′n矩阵,B是n′m矩阵,则

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《线性代数》英文专业词汇 序号英文中文1Linear Algebra线性代数 2determinant行列式 3row行 4column列 5element元素 6diagonal对角线 7principal diagona主对角线 8auxiliary diagonal次对角线 9transposed determinant转置行列式 10triangular determinants三角行列式 11the number of inversions逆序数 12even permutation奇排列 13odd permutation偶排列 14parity奇偶性 15interchange互换 16absolute value绝对值 17identity恒等式 18n-order determinants n 阶行列式 19evaluation of determinant行列式的求值 20Laplace ’s expansion theorem拉普拉斯展开定理21cofactor余子式 22Algebra cofactor代数余子式 23the Vandermonde determinant范德蒙行列式 24bordered determinant加边行列式 25reduction of the order of a determinant降阶法 26method of Recursion relation递推法 27induction归纳法 28Cramer′s rule克莱姆法则 29matrix矩阵 30rectangular矩形的 31the zero matrix零矩阵

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Chapter 4 Linear Transformations In this chapter, we introduce the general concept of linear transformation from a vector space into a vector space. But, we mainly focus on linear transformations from n R to m R. §1 Definition and Examples New words and phrases Mapping 映射 Linear transformation 线性变换 Linear operator 线性算子 Dilation 扩张 Contraction 收缩 Projection 投影 Reflection 反射 Counterclockwise direction 反时针方向 Clockwise direction 顺时针方向 Image 像 Kernel 核 1.1 Definition ★Definition A mapping(映射) L: V W is a rule that produces a correspondence between two sets of elements such that to each element in the first set there corresponds one and only one element in the second set. ★Definition A mapping L from a vector space V into a vector space W is said to be a linear transformation（线性变换）if

线代名词中英文对照

《线性代数》英文专业词汇 序号英文中文 1LinearAlgebra线性代数 2determinant行列式 3row行 4column列 5element元素 6diagonal对角线 7principaldiagona主对角线 8auxiliarydiagonal次对角线 9transposeddeterminant转置行列式 10triangulardeterminants三角行列式 11thenumberofinversions逆序数 12evenpermutation奇排列 13oddpermutation偶排列 14parity奇偶性 15interchange互换 16absolutevalue绝对值 17identity恒等式 18n-orderdeterminantsn阶行列式 19evaluationofdeterminant行列式的求值 20Laplace’sexpansiontheorem拉普拉斯展开定理 21cofactor余子式 22Algebracofactor代数余子式 23theVandermondedeterminant范德蒙行列式 24bordereddeterminant加边行列式 25reductionoftheorderofadeterminant降阶法 26methodofRecursionrelation递推法 27induction归纳法 28Cramer′s rule克莱姆法则 29matrix矩阵 30rectangular矩形的 31thezeromatrix零矩阵

32theidentitymatrix单位矩阵 33symmetric对称的 序号英文中文 34skew-symmetric反对称的 35commutativelaw交换律 36squareMatrix方阵 37amatrixoforder m×n矩阵m×n 38thedeterminantofmatrixA方阵A的行列式39operationsonMatrices矩阵的运算 40atransposedmatrix转置矩阵 41aninversematrix逆矩阵 42anconjugatematrix共轭矩阵 43andiagonalmatrix对角矩阵 44anadjointmatrix伴随矩阵 45singularmatrix奇异矩阵 46nonsingularmatrix非奇异矩阵 47elementarytransformations初等变换 48vectors向量 49components分量 50linearlycombination线性组合 51spaceofarithmeticalvectors向量空间 52subspace子空间 53dimension维 54basis基 55canonicalbasis规范基 56coordinates坐标 57decomposition分解 58transformationmatrix过渡矩阵 59linearlyindependent线性无关 60linearlydependent线性相关 61theminorofthe k thorderk阶子式 62rankofaMatrix矩阵的秩 63rowvectors行向量

完整word版线性代数英文专业词汇x

《线性代数》英文专业词汇序号英文中文

1Linear Algebra线性代2determinant行列3row 4column5element元6diagonal对角7principal diagona主对角8auxiliary diagonal次对角9 transposed determinant转置行列10triangular determinants三角行列11the number of inversions逆序12even permutation奇排13odd permutation偶排14parity奇偶15interchange 互16absolute value绝对17identity恒等18 n-order determinants n 阶行列式 19evaluation of determinant行列式的求20 Laplace 's expansion theorem拉普拉斯展开21 cofactor余子22Algebra cofactor代数余子式23the Vandermonde determinant范德蒙行列24 bordered determinant加边行列25reduction of the order of a determinant降阶26method of Recursion relation递推27induction归纳28 Cramer′s rule克莱姆法29matrix矩30 rectangular矩形31the zero matrix零矩阵

32the identity matrix单位矩33symmetric对称的 序号英文中文 34skew-symmetric反对称35commutative law

自考04184线性代数(经管类)讲义

自考高数线性代数课堂笔记 第一章行列式 线性代数学的核心内容是：研究线性方程组的解的存在条件、解的结构以及解的求法。所用的基本工具是矩阵，而行列式是研究矩阵的很有效的工具之一。行列式作为一种数学工具不但在本课程中极其重要，而且在其他数学学科、乃至在其他许多学科（例如计算机科学、经济学、管理学等）都是必不可少的。 1.1行列式的定义 （一）一阶、二阶、三阶行列式的定义 （1）定义：符号叫一阶行列式，它是一个数，其大小规定为：。 注意：在线性代数中，符号不是绝对值。 例如，且； （2）定义：符号叫二阶行列式，它也是一个数，其大小规定为： 所以二阶行列式的值等于两个对角线上的数的积之差。（主对角线减次对角线的乘积）例如 （3）符号叫三阶行列式，它也是一个数，其大小规定为 例如=0 三阶行列式的计算比较复杂，为了帮助大家掌握三阶行列式的计算公式，我们可以采用下面的对角线法记忆 方法是：在已给行列式右边添加已给行列式的第一列、第二列。我们把行列式左上角到右下角的对角

线叫主对角线，把右上角到左下角的对角线叫次对角线，这时，三阶行列式的值等于主对角线的三个数的积与和主对角线平行的线上的三个数的积之和减去次对角线三个数的积与次对角线的平行线上数的积之和。 例如： （1） =1×5×9+2×6×7+3×4×8-3×5×7-1×6×8-2×4×9=0 （2） （3） （2）和（3）叫三角形行列式，其中（2）叫上三角形行列式，（3）叫下三角形行列式，由（2）（3）可见，在三阶行列式中，三角形行列式的值为主对角线的三个数之积，其余五项都是0，例如 例1a为何值时，

[答疑编号10010101：针对该题提问] 解因为 所以8-3a=0，时 例2当x取何值时， [答疑编号10010102：针对该题提问] 解： 解得0

数学专业名词英文解释

高等数学(Higher Mathematics)一些基本名词中英文对照表 中文英文中文英文 counterclockwise 函数function 逆时针方 向 定义域domain of definition 变量variable 值域range of function 常量constant quantity 极限limit 坐标轴axis of coordinates 极限值limit value 横坐标abscissa 发散diverge 纵坐标ordinate 收敛converge 锐角acute angle 连续性continuity 钝角obtuse angle 连续函数continuous function 平角straight angle 左连续continuity from the left 直角right angle 开集open set 圆circle 闭集closed set 半径radius 闭区间closed interval 直径diameter 区间interval 三角形triangle 一元函数function of one variable 斜率slope 多元函数function of several variables 无穷小infinitesimal 内点inner point 无穷大infinite 孤立点isolated point 正positive 邻域neighborhood 负negative 导数derivative 凸convex 偏导数partial derivative 凹concave 微分differential calculus 椭圆ellipse 全微分total differential 双曲线hyperbola 偏微分partial differential 曲线curve 积分integral 曲面surface 微积分infinitesimal calculus 交intersection 重积分multiple integral 补集complement 二重积分double integral 投影project

线性代数英文词汇

英汉词汇（按英文字母排序） A adjont(adjugate) of matrix A A 的伴随矩阵 augmented matrix A 的增广矩阵 B block diagonal matrix 块对角矩阵 block matrix 块矩阵 basic solution set 基础解系

C Cauchy-Schwarz inequality 柯西 - 许瓦兹不等式characteristic equation 特征方程 characteristic polynomial 特征多项式 coffcient matrix 系数矩阵 cofactor 代数余子式 cofactor expansion 代数余子式展开 column vector 列向量 commuting matrices 交换矩阵 consistent linear system 相容线性方程组 Cramer's rule 克莱姆法则 Cross- product term 交叉项 D Determinant 行列式 Diagonal entries 对角元素 Diagonal matrix 对角矩阵 Dimension of a vector space V 向量空间V 的维数E echelon matrix 梯形矩阵 eigenspace 特征空间 eigenvalue 特征值 eigenvector 特征向量 eigenvector basis 特征向量的基 elementary matrix 初等矩阵

elementary row operations 行初等变换 F full rank 满秩 fundermental set of solution 基础解系 G grneral solution 通解 Gram-Schmidt process 施密特正交化过程 H homogeneous linear equations 齐次线性方程组I identity matrix 单位矩阵 inconsistent linear system 不相容线性方程组indefinite matrix 不定矩阵 indefinit quatratic form 不定二次型 infinite-dimensional space 无限维空间 inner product 内积

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Student Number: Your Name: 2016—2017 Fall Semester UNIVERSITY OF SCIENCE ＆ TECHNOLOGY BEIJING Linear Algebra Final Exam Time:09:00-11:30 A.M.Full Mark:100 Your Mark: Notation: Please fill out and sign the front of your exam booklet. No books or electronic devices allowed. No using any notes or formulas! No cheating! You may keep this paper. Solutions will be posted on the course website after the exam. Please do not answer the following problem until we give the signal. 1. (20 points) Let A =2 2 1 6 6 3 ?3 1 10 4 13 10 4 Find bases of the following vector spaces and statetheir dimensions. (a) The column space of A. (b) The row space of A. (c) The null space of A. (d) The orthogonal complement of the column space of A. 2. (15 points)Let A = 4?1 30 (a) Compute A k for all integers k≥0. Write the answer as explicitly as you can, in the form ofa 2×2-matrix with entries depending on k. (b) Solve the initial value problem x'(t) = A x(t) with x(0) =1 0 3. (10 points) (a) Let P n be the vector space of polynomials of degree less than or equal to n.Let T be the linear transformation from P3to P4defined by T(p)(t)=p(2) + (t– 2)p (t)+ t3p (5t) (You are not required to show that T is linear.) Find the matrix of T with respect to the B 3= {1, t, t2, t3} of P3 and the B 4 = {1,t,t2,t3,t4} of.P4 (b) Find the equation y = ax + b of the least-squares line that best fits the data points (1,2), (2,2), and (3,4).

大学线性代数必过复习资料

复习重点： 第一部分行列式 1．排列的逆序数（P.5 例4；P.26 第2、4 题） 2．行列式按行（列）展开法则（P.21 例13；P.28 第9 题）3．行列式的性质及行列式的计算（P.27 第8 题） 第二部分矩阵 1．矩阵的运算性质 2．矩阵求逆及矩阵方程的求解（P.56 第17、18 题；P.78 第5 题） 3．伴随阵的性质（P.41 例9；P.56 第23、24 题；P.109 第 25 题）、正交阵的性质（P.116） 4．矩阵的秩的性质（P.69 至71；P.100 例13、14、15） 第三部分线性方程组 1．线性方程组的解的判定（P.71 定理3；P.77 定理4、5、6、7），带参数的方程组的解的判定（P.75 例13；P.80 第16、 17、18 题） 2．齐次线性方程组的解的结构（基础解系与通解的关系） 3．非齐次线性方程组的解的结构（通解）

第四部分向量组（矩阵、方程组、向量组三者之间可以相互转换） 1 .向量组的线性表示 2.向量组的线性相关性 3.向量组的秩 第五部分方阵的特征值及特征向量 1.施密特正交化过程 2.特征值、特征向量的性质及计算（P.120例8 9、10; P.135 第7至13题） 3.矩阵的相似对角化，尤其是对称阵的相似对角化（P.135第15、16、19、23 题） 要注意的知识点： 线性代数 1、行列式 1. n行列式共有n2个元素，展开后有n!项，可分解为2n行列式； 2?代数余子式的性质： ①、A j和a,的大小无关； ②、某行（列）的元素乘以其它行（列）元素的代数余子式 为0； ③、某行（列）的元素乘以该行（列）元素的代数余子式为A ；

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高等数学(Higher Mathematics) 一些基本名词中英文对照表 中文英文中文英文 函数 function 逆时针方 向 counterclockwise 定义域domain of definition 变量variable 值域 range of function 常量 constant quantity 极限 limit 坐标轴 axis of coordinates 极限值 limit value 横坐标 abscissa 发散 diverge 纵坐标 ordinate 收敛 converge 锐角 acute angle 连续性 continuity 钝角 obtuse angle 连续函数 continuous function 平角 straight angle 左连续 continuity from the left 直角 right angle 开集 open set 圆 circle 闭集 closed set 半径 radius 闭区间 closed interval 直径 diameter 区间 interval 三角形 triangle 一元函数 function of one variable 斜率 slope 多元函数 function of several variables 无穷小 infinitesimal 点 八、、inner point 无穷大 infinite 孤立点 isolated point 正 positive 邻域 neighborhood 负 negative 导数 derivative 凸 convex 偏导数 partial derivative 凹 concave 微分 differential calculus 椭圆 ellipse 全微分 total differential 双曲线 hyperbola 偏微分 partial differential 曲线 curve 积分 integral 曲面 surface 微积分 infinitesimal calculus 交 intersection 重积分 multiple integral 补集 complement 二重积分double integral 投影project

2014汤家凤线性代数辅导讲义

文都教育2014年考研数学春季基础班线性代数辅导讲义 主讲：汤家凤 第一讲 行列式 一、基本概念 定义1 逆序—设j i ,是一对不等的正整数，若j i >，则称),(j i 为一对逆序。 定义2 逆序数—设n i i i 21是n ,,2,1 的一个排列，该排列所含逆序总数称为该排列的逆序数，记为)(21n i i i τ，逆序数为奇数的排列称为奇排列，逆序数为偶数的排列称为偶排列。 定义3 行列式—称nn n n n n a a a a a a a a a D 21 2222111211 = 称为n 阶行列式，规定 n n n nj j j j j j j j j a a a D 21212121) ()1(∑-= τ 。 定义4 余子式与代数余子式—把行列式nn n n n n a a a a a a a a a D 21 22221112 11 = 中元素ij a 所在的i 行元 素和j 列元素去掉，剩下的1-n 行和1-n 列元素按照元素原来的排列次序构成的1-n 阶行列式，称为元素ij a 的余子式，记为ij M ，称ij j i ij M A +-=)1(为元素ij a 的代数余子式。 二、几个特殊的高阶行列式 1、对角行列式—形如 n a a a 000 0021 称为对角行列式， n n a a a a a a 2121 00 00 0=。 2、上（下）三角行列式—称 nn n n a a a a a a 00 022211211及 nn n n a a a a a a 21 222111 0为上（下）三角行

列式， nn nn n n a a a a a a a a a 221122211211 00 0=， nn nn n n a a a a a a a a a 221121222111 0=。 3、 ||||B A B O O A ?=， ||||B A B O C A ?=， ||||B A B C O A ?=。 4、范得蒙行列式—形如1121 121211 11 ),,,(---= n n n n n n a a a a a a a a a V 称为n 阶范得蒙行列式，且 n i j j i n n n n n n a a a a a a a a a a a V ≤<≤----== 11121 12 121)(1 11 ),,,(。 【注解】0),,,(21≠n a a a V 的充分必要条件是n a a a ,,,21 两两不等。 三、行列式的计算性质 （一）把行列式转化为特殊行列式的性质 1、行列式与其转置行列式相等，即T D D =。 2、对调两行（或列）行列式改变符号。 3、行列式某行（或列）有公因子可以提取到行列式的外面。 推论1行列式某行（或列）元素全为零，则该行列式为零。 推论2行列式某两行（或列）相同，行列式为零。 推论3行列式某两行（或列）元素对应成比例，行列式为零。 4、行列式的某行（或列）的每个元素皆为两数之和时，行列式可分解为两个行列式，即 nn n n in i i n nn n n in i i n nn n n in in i i i i n a a a b b b a a a a a a a a a a a a a a a b a b a b a a a a 2 121112112 121112 11212 21 111211 +=+++。