Linear Algebra with 



For ,

express the vectors  in component form.




        and  are equivalent.


Copy the following code into http://sage.skku.edu

                or http://mathlab.knou.ac.kr:8080/ to practice.

For vectors  in  , find , and .

● http://matrix.skku.ac.kr/RPG_English/1-VT-sum-multi.html 

Copy the following code into

http://sage.skku.edu or http://mathlab.knou.ac.kr:8080/ to practice.

For ,

, express the vectors 

in component form.



and  are equivalent. 

Copy the following code into http://sage.skku.edu

                or http://mathlab.knou.ac.kr:8080/ to practice.



 Find  when  and  in .

● http://matrix.skku.ac.kr/RPG_English/1-VT-sum-multi-3.html 



Find , when  and in .



Copy the following code into http://sage.skku.edu or

             http://mathlab.knou.ac.kr:8080/ to practice.





  The above  can also be done in Sage as follows. First, we build the relevant vectors and the command for a linear combination of many vectors. Then, we can combine all into one line, as follows.   


For the vectors   in , we have the following.

● http://matrix.skku.ac.kr/RPG_English/1-B1-norm-distance.html 



Copy the following code into http://sage.skku.edu to practice.

Using the vectors  and  in , calculate .




Copy the following code into http://sage.skku.edu to practice.



For two vectors  and  in , establish orthogonality.


Find Angle   (두 벡터의 사잇각을 구하여라)

 For vectors  calculate , where   is

the angle between x and y.





Note : ∴ 


Checked by Sage)  http://math1.skku.ac.kr/home/pub/2508/ by 임상훈

2015.9.9. Solved by 김성찬 2015.9.9. Finalized by 김성찬 2015.9.13. Refinalized by 정다산 2015.9.23. Refinalized by 김성찬, 임상훈

2015.9.29. Final OK by 이경승 Final OK by SGLe

Using the vectors  and  from , verify that the triangle inequality holds.




. Hence


So, .                      



Section 3 : Projection / Vector Equation  (정사영, 직선과 평면의 벡터 방정식)


Find vector, parametric and symmetric equations of the line that passes through the point  and is parallel to the vector .

(1) The vector equation of the line is given by

            ,    or


(2) The parametric equation is given by 


(3) The symmetric equation is given by .   


Find parametric equations for the line that passes through the points  and .


Two points  and  with position vectors  and  forms a vector




  and the vector equation  can be written  as




 Thus, the parametric equations are:



Find vector and parametric equations of the plane that passes through the three points:  , and .


● http://matrix.skku.ac.kr/RPG_English/1-BN-11.html 


Let , and . Then we have two vectors that parallel to the plane as




Then, from our above definitions, we have


which is a vector equation of the plane.


If we further simplify the above expression, we have



 In particular,     

is the parametric equations of the plane.                         ■


For vectors , find  (the projection of  onto and  (the component of orthogonal to ).

   ( Find p (the projection of y onto x) and w = y-p )

Since , we have


Example 5: Find the distance D. 

HW : Find your own problem in R^5 and practice it in the following box.


HW (과제) : Textbook EXS of Chapter 1 :

Even numbers,  (Example) 2, 4, 6, 8, … , P2, P4, …  

                  (짝수번 문제들)

    Solve, Change the problem, Solve, Share your 2 or 3 problems and solution in QnA.

   You will be asked what you shared in QnA regularly.

                            (여러분의 답을 QnA 에 업로드 하시면 됩니다.) )