linerar system theory and design 线性系统理论陈启宗答案

3.1consider Fig3.1 ,what is the representation of the vector with respect to the basis

? What is the representation of with respect ro ?图 3.1 中,向量 关于

的表示是什么? 关于 的表示有是什么?

If we draw from two lines in parallel with and , they tutersect at and as

shown , thus the representation of with respect to the basis is , this can

be verified from

To find the representation of with respect to ,we draw from two lines in

parallel with and , they intersect at -2 and , thus the representation of with

The two orthomormal vectors are

In fact , the vectors and are orthogonal because so we can only

normalize the two vectors

3.4 comsider an matrix A with , if all colums of A are orthonormal , then

, what can you say abort ? 一个 阶矩阵 A( ), 如果 A 的所有列都

是标准正交的,则 问 是怎么样?

Let , if all colums of A are orthomormal , that is

then

if A is a symmetric square matrix , that is to say , n=m for every

then

linearly independent , the third colums can be expressed as , so the set

can be used as a basis of the range space of

3.7 consider the linear algebraic equation it has three equations and two

unknowns does a solution exist in the equation ? is the solution unique ? does a solution exist if

?

线性代数方程 有三个方程两个未知数, 问方程是否有解?若有解是否

唯一?若 方程是否有解?

Let clearly and are linearly independent , so rank(A)=2 ,

is the sum of and ,that is rank([A ])=2, rank(A)= rank([A ]) so a solution

exists in A =

lies in the range space of A and is a solution Nullity(A)=4-3=1 that

means the dimension of the null space of A is 1 , the number of parameters in the general solution

will be 1 , A basis of the null space of A is thus the general solution of A

= can be expressed an for any real

is the only parameter

3.9 find the solution in example 3.3 that has the smallest Euclidean norm 求例 3 中具有最小欧氏

范数的解,

the general solution in example 3.3 is for any

real and

the Euclidean norm of is

where A is an matrix and is an column vector ,under what conditions on A and will there

exist to meet the equation for any ?

令 A 是 的矩阵, 是 的列向量,问在 A 和 满足什么条件时,存在 ,对所有

的 ,它们都满足方程

write the equation in this form

where is an matrix and is an column vector , from the equation

we can see , exist to meet the equation for any ,if and only if

any .