LA Chapter 9 by SGLee


Chapter 9

Vector Space


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9.1 Axioms of a Vector Space

9.2 Inner product; *Fourier series

9.3 Isomorphism

9.4 Exercises


The operations used in vector addition and scalar multiple are not limited to the theory but can be applied to all areas in society.

For example, consider objects around you as vectors and make a set of vectors,

then create two proper operations (vector addition and scalar multiple) from the relations between the objects.

 If these two operations satisfy the two basic laws and 8 operation properties, the set becomes a mathematical vector space (or linear space).

   Thus we can use all properties of a vector space and can analyze set theoretically and apply them to real problems. 


In this chapter, we give an abstract definition of a vector space and deal with general theory of a vector space.

9.1 Axioms of a Vector Space


 Ref site :

 Lab site: 

In this section. concept of vectors has been extended to -tuples in  from the arrows in the 2-dimensional or 3-dimensional space.

In Chapter 1, we defined the addition and the scalar multiplication in the -dimensional space .

  In this section, we extend the concept of the -dimensional space  to an -dimensional vector space.

Vector Spaces



 [Vector space]





If a set  has two well-defined binary operations, vector addition (A) ‘’ and scalar multiplication (SM) ‘’, and for any  and , two basic laws


  A. .

  SM. .


and the following eight laws hold, then we say that the set  forms a

vector space over  with the given two operations, and we denote it by (simply  if there is no confusion). Elements of  are called vectors.


  A1. .

  A2. .

  A3.For any , there exists a unique element  in such that


  A4. For each element  of , there exists a unique  such that



  SM1. .

  SM2. .

  SM3. .

  SM4. .


The vector  satisfying A3 is called a zero vector, and the vector satisfying A4 is called a negative vector of .








 In general, the two operations defining a vector space are important. Therefore, it is better to write  instead of just .



For vectors  in  and a scalar , the vector sum and a scalar multiple  by  are defined as


(1) .

(2)  .

The set  together with the above operations forms a vector space over the set  of real numbers.                   



For vectors in 




and a scalar , the sum of two vectors  and the scalar multiple of  by  is defined by


            (1)   and  (2) .


The set  form a vector space together with the above two operations. 








Let be a vector space. Let  and . Then the following hold.

(1) .

(2) .

(3) .

(4)  or .


Zero Vector Space







Let . For a scalar , if the addition and scalar multiple are defined as

                        , then


 forms a vector space. This vector space is called a zero vector space.







Let  be the set of all  matrices with real entries. That is,




When , we denote   by  .


If  is equipped with the (usual) matrix addition and the scalar multiplication,

then  form a vector space  over .

 The zero vector is the zero matrix  and for each , the negative vector is .

  Note that each vector in  means an  matrix with real entries.                            



Let  be the set of all continuous functions from  to . That is,


                    is continuous}


Let  and a scalar , define the addition and the scalar multiple as




Then  forms a vector space  over .


The zero vector is  (zero function) and for each  is defined as .


Vectors in  mean continuous functions from  to .    



Let  be the set of all polynomials of degree at most  with real coefficients. In other words,




Let  and a scalar .

     The addition and the scalar multiplication are defined as





Then  forms a vector space  over .

     The zero vector is  zero polynomial and each  has the negative vector defined as




Vectors in  means polynomials of degree at most  with real coefficients.■










Let  be a vector space and  be a subset of .

   If  forms a vector space with the operations defined in , then  is called a subspace of .








If  is a vector space,  and  itself are subspaces of  called the trivial subspaces.     



 In fact, the only subspaces of  are , and lines passing through the origin. (see section 3.4 ).


 In , only subspaces are

     (i)         (Null Space),

     (ii)    ,

    (iii)     lines passing through origin and (iv) planes passing through origin.


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 How to determine a subspace?




 9.1.2 [the 2-step subspace test]

Let a set  be a vector space and  be a subset of .

A necessary and sufficient condition for  to be a subspace of  is 


(1) (closed under vector addition )

(2) (closed under scalar multiple)


Show that  is a subspace of the vector space .


Note that  is a vector space under the matrix addition and the scalar multiplication. Let


The following two conditions are satisfied.



         (2) .


Hence by Theorem 9.1.2,  is a subspace of .  

The set of invertible matrices of order  is not a subspace of the vector space .


One can make a non-invertible matrix by adding two invertible matrices. For example,




Let  be a vector space and . Show that the set




is a subspace of . Note that  is a linear span of the set .


Suppose that . Then for ,









          and .


Therefore,  is a subspace of .                               




Linear independence and linear dependence


 [Linear independence and linear dependence]





If a subset  of a vector space  satisfies the following condition, it is called linearly independent.




and if the set is not linearly independent, it is called linearly dependent.

Hence being linearly independent means that

there exist some scalars  not all zero such that .








 Linear combination in 2-dimensional space - linear dependence  

 (computer simulation)















Let . Since




 is a linearly independent set of .      

Let .

 Since  is a linearly dependent set of .    



The subset  of  is linearly independent.     




Let  be a subset of . Then since




the set is linearly dependent.                                    







 [basis and dimension]





If a subset () of a vector space  satisfies the following conditions,  is a basisof .


(1) .

(2)  is linearly independent.


In this case, the number of elements of the basis , is called the dimension of , denoted by 








        The set in  consisting of  is a basis of .


              Thus .

      On the other hand, the set in   is a basis of .    Thus .


       These bases play a role similar to the standard basis of , hence

} and  are called standard bases for  and respectively.         



Show that  is a basis of .





Since  is linearly independent.


Next, given , the existence of  such that




 is guaranteed since the coefficient matrix of the linear system


         that is, 


is invertible. Thus  spans . Hence  is a basis of .   


Linear independence of continuous function:





 9.1.3 [Wronski's Test]

If  are  times differentiable on the interval and

there exists  such that Wronskian  defined below is not zero,

then these functions are linearly independent.




Conversely if  for every  in , then  are linearly dependent.


Show by Theorem 9.1.3 that  are linearly independent.


    For some (in fact, any) .


         Thus these functions are linearly independent.                          

2*e^(3*x)                                                         ■

Let . Show that these functions are linearly independent.


Since  for some ,

these functions are linearly independent.               


Show that  are linearly dependent.


Since for any ,



these functions are linearly dependent.        



9.2 Inner product; *Fourier series


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In this section, we generalize the Euclidean inner product on  (dot product) to introduce

the concepts of length, distance, and orthogonality in a general vector space.

Inner product and inner product space


 [Inner product and inner product space]





The inner product on a real vector space  is a function assigning a pair of vectors ,

to a scalar  satisfying the following conditions.

  (that is, the function  satisfies the following conditions.)


(1)  for every  in .

(2)  for every  in .

(3)  for every  in  and  in .

(4)  for every  in .


The inner product space is a vector space  with an inner product  defined on .







The Euclidean inner product, that is, the dot product is an example of an inner product on .

  Let us ask how other inner products on  are possible. For this, consider .

   Let  and  be the column vectors of . Define 

                                (or by .

Then let us find the condition on  so that this function becomes an inner product.


In order for  to be an inner product, the four conditions (1)~(4) should be satisfied. First consider conditions (2) and (3).       








Let us check when condition (1) holds. Since  is a  matrix (hence a real number), we have




That is, to satisfy




we must have , in other words,  is a symmetric matrix.


Thus the function  satisfy condition (1) if  is a symmetric matrix.


Finally check condition (4). An  symmetric matrix  should satisfy  for any nonzero vector .

  This condition means that  is positive definite. In other words, if  is positive definite,  satisfies condition (4).


Therefore, to wrap up, if  is an  symmetric and positive definite matrix,

  then  defines an inner product on .

   The well known Euclidean inner product  can be obtained as a special case

   when  (symmetric and positive definite).       


 For any nonzero vector , if the eigenvalues of  are positive, then    (the converse also holds.)



Let  be a  symmetric matrix and  in . Then



satisfies conditions (1), (2), (3) of an inner product on .

Now let us show that  is a positive definite. Let . Then


   . Thus





Hence the symmetric matrix  is positive definite and

       defines an inner product on  of the form .


If  and , then . On the other hand,



Hence the inner product  on  is different from the Euclidean inner product.                      

Norm and angle




 [norm and angle]





Let  be a vector space with an inner product . The norm (or length) of a vector  with respect to the inner product is defined by



The angle  between two nonzero vectors  and  is defined by




In particular, if two vectors  and  satisfy , then they are said to beorthogonal.








 For example, the norm of  with respect to the inner product given in  is



Thus . On the other hand, the norm  with respect to the Euclidean inner product is






 For any inner product space, the triangle inequality  holds.


 Using the Gram-Schmidt orthogonality process,

we can make a basis  of a inner product space  into an orthonormal basis .



Inner product on complex vector space









Let  be a complex vector space. Let  be any vectors in  and  be any scalar. The function  from   to  is called an inner product (orHermitian inner product) if the following hold.

(1) .

(2) .

(3) .








 A complex vector space with an inner product is called a complex inner product space or a unitary space

 If  for any two nonzero vectors , then we say that  and  are orthogonal.


 Let  be a complex vector space. By the definition of an inner product on , we obtain the following properties.


  (1) .

  (2) .



Let  and  be vectors in .

  The Euclidean inner product  satisfies the conditions  (1)~(4) for the inner product.            

Let  be the set of continuous functions from the interval  to the complex set .

   Let . If the addition and scalar multiple of these functions are defined below,

    then  is a complex vector space with respect to these operations.




In this case, a vector in  is of the form and are continuous functions from  to .

    For  , define the following inner product




Then  is a complex inner product space.


We leave readers to check conditions (1)~(3) for an inner product, and show condition (4) here. Note




and , hence . In particular,




That is, , conversely, if  is a zero function, then it is easy to see that .           

Complex inner product space, norm, distance




[Norm, and distance]





Let  be a complex inner product space. The norm of  and the distance between  and  are defined as follows:









Find the Euclidean inner product and the distance of vectors  .










From , we let  and . Find the norm of .





Cauchy-Schwarz inequality and the triangle







Let  be a complex inner product space. For any  in , the following hold.


(1) .      (Cauchy-Schwarz inequality)

(2) .       (triangle inequality)



 We prove (1) only and leave the proof of (2) as an exercise.

If . Hence (1) holds. Let  and  .

 Then  and . Thus we have the following.









       Thus, as, (1) holds.   



Let  be vectors in . Answer the following.


(1) Compute the Euclidean inner product .

(2) Show that  and  are linearly independent.


(1) .






(2) If  for any scalar , then




   So . Thus  and  are linearly independent.      

Let  be vectors in .

   Check that the Cauchy-Schwarz inequality and the triangle inequality hold.


Since  and , the Cauchy-Schwarz inequality holds.

Also since , the triangle inequality holds.                             





[Cauchy-Schwarz inequality in  and ]

(1) Let  be a complex inner product space with the Euclidean inner product.

   Let  be in Then the Cauchy-Schwarz inequality is given by




(2) Let . As in with the inner product, the Cauchy-Schwarz inequality is given by







[Triangle inequality] Consider the inner products given in  and .


(1) Let . Then the triangle inequality holds. That is,




(2) Let . Then the triangle inequality holds. That is,







9.3 Isomorphism


 Reference site: 

 Lab site: 

We generalize the definition of a linear transformation on  to a general vector space . 

A special attention will be given to both injective and surjective linear transformations. 









Let  and  be vector spaces over  be a map from a vector space to a vector space .

If  satisfies the following conditions, it is called a linear transformation.


(1)                 for every  in  and  in .

(2)     for every  in .









 If , then the linear transformation  is called a linear operator.





If  is a linear transformation, Then we have the following:


(1) .

(2) .

(3) .





If  satisfies that  for any , then it is a linear transformation,

    called the zero transformation. Also, if  satisfies that  for any ,

    then it is a linear transformation, called the identity operator.             



Define  by  ( a scalar). Then  is a linear transformation. The following two properties hold.





If  , then  is called a contraction and if , then it is called adilation.                               



Let  be the vector space of all continuous functions from  to  and  be the subspace of 

     consisting of differentiable functions.

  Define  by . Then  is a linear transformation and called aderivative operator.           




Let  the subspace of  consisting of differentiable functions.

Define  by . Then  is linear transformation.    



Kernel and Range




 [Kernel and Range]





Let . Define




ker is called the kernel and Im the range







If  is the zero transformation,  and .  



If  is the identity operator,  and .     



Let  be the derivative operator defined by  as in .

      is “the set of all constant functions defined on ” and

      is “the set of all continuous functions, that is, ”       

Basic properties of kernel and range





If  is a linear transformation,  and  are subspaces of and  respectively.






If  is a linear transformation, the following statements are equivalent.


(1)  is an injective (or one-to-one) function.

(2) .














If a linear transformation  is one-to-one and onto,

   then it is called an isomorphism. In this case, we say that  is isomorphic to ,

   denoted by .







   Any -dimensional real vector space (defined over the real set ) is isomorphic to and

       any -dimensional complex vector space (defined over the complex set ) is isomorphic to .




Any -dimensional real vector space is isomorphic to .



We immediately obtain the following result from the above theorem.



Ch. 9 Exercises



9.1 Axioms of a Vector Space  (벡터공간의 공리)

9.2 Inner product; *Fourier series  (내적공간; *푸리에 급수)

9.3 Isomorphism  (동형사상)


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