Relevance Research and Teaching Practice of
Linear Algebra Course Content
Rui Chen 1* and Liang Fang 2
College of Mathematics and Statistics, Taishan University, 271000, Tai'an, Shandong, China.
Date of publication (dd/mm/yyyy): 26/03/2020
Abstract – Linear Algebra is an important instrumental course, and its large number of concepts and complicated operations make it more difficult for students to learn. Starting from the characteristics of the course, this paper focuses on the overall structure of the course, and Constructs Knowledge Association, therefore, the coherence and systematicness of teaching and learning will be enhanced, the students' comprehension ability will be improved and their thinking will be expanded with the help of practical examples, and the teaching quality and efficiency of Linear Algebra will be improved effectively.
Keywords – Linear Algebra, Relevance, Wholeness, Knowledge Structure, Teaching Practice.
I.
I
NTRODUCTIONAccording to the characteristics of the course, we can find out the logical venation of the knowledge content, dredge the relation between the concepts, and integrate the scattered knowledge points, it is an important way to promote students’ understanding of the course, to cultivate their good logical thinking ability, and to analyze and solve problems [1]. Many teachers have studied the teaching practice of Linear Algebra, it is pointed out that the course of Linear Algebra has strong abstractness and complexity, rich content, strong internal connection and great difficulty in learning [2]. Based on the characteristics of the Linear Algebra course and the actual situation of the students, some teachers put forward that the teaching of Linear Algebra should be combined with concrete examples to arouse the students’ interest in learning, and the use of examples or counterexamples to enable students to understand the concept, so as to grasp its connotation and operation, to really understand the course objectives [3].
One of the characteristics of Linear Algebra is that there are many concepts, many property theorems, many operation rules, and a large amount of operation, the main content of Linear Algebra includes five parts: determinant, Matrix, system of linear equations, Linear Independence of vector group, similar matrix and quadratic form, the tools it uses are matrices and determinants, and these seemingly unrelated elements have a one to one correspondence in their forms [4]: Determinants and system of linear equations, matrices and system of linear equations, matrices and vector groups, in teaching, the teacher may take one of the contents as the starting point of the course content, construct the connection with other branches, and advance step by step, thus forms the system knowledge system, the solution abstract concept many difficult problems.
II.
P
ARTIALK
NOWLEDGEC
ONSTRUCTION OFL
INEARA
LGEBRAC
OURSE2.1. Constructing the Knowledge Thread between the System of Linear Equations, the Determinant and the Cramer Rule
System of linear equations is the core concept and important research object of Linear Algebra. From the simple elimination method of system of linear equations, we introduce the second order determinant and the third order determinant, by analyzing the characteristics of the third order determinant, the definition of higher
order determinant is introduced. The calculation of determinant is not the main content of the course, but as the foreshadowing of square matrix correlation operation or the prior knowledge of Cramer rule, it needs to be explained as a piece of independent content. Cramer's rule solves the problem of the partial solution of the same number of equations and unknowns as the system of linear equations [5]. We show it by an example.
Example 1: Solving a system of linear equations
1 2 3 4 2 3 4 1 3 4 1 2 3 2 3, 2 1, 2 2, 3 2 5. x x x x x x x x x x x x x (1)
Method 1: First, write the determinant of the coefficients of the system of linear equations
1 1 1 2 0 1 2 1 . 1 0 1 2 1 3 2 0 D
Then, applying the properties of the determinant, the value of the determinant is 15. The Cramer rule states that when the value of the determinant of the Coefficient is not equal to 0, the system of linear equations has a unique solution. 3 1 2 4 1 2 3 4 22 31 20 4 6 2 , , , . 15 15 15 3 15 5 D D D D x x x x D D D D
2.2. Constructing the Knowledge Thread between System of Linear Equations and Matrix Operations
The GAUSS elimination method of the system of linear equations is an effective method for solving equations that students are familiar with. If the coefficients and constant terms of the unknown variables in the system of linear equations are kept in the same order as the original position, they are arranged into a table of numbers, by reviewing the system of linear equations method, we can lead the students to discover that the process of solving equations is in fact the process of the same solution deforming, this process involves the operation of unknown coefficients and constant terms, corresponding to the three row transformations of the Matrix, namely, the transformation of Pairs, the transformation of multiplication, the transformation of multiplication, and the transformation of multiplication, which is the Elementary matrix of the Matrix, with the Elementary Matrix, we call it the elementary transformation of the Matrix. Through the Elementary Matrix of the Matrix, we can transform the matrix into a Row echelon form, and then into a row-simplest Matrix, the Matrix elementary change method for solving the system of linear equations problem can be used to solve the inverse Matrix and rank of the Matrix [2]. Starting from the rule of Matrix multiplication, system of linear equations can be abbreviated asAx ,A is the Coefficient Matrix of the equation system,
x
is an unknown column vector and is a constant column vector. IfA
is reversible, you can get the unique solution ofAx to bex A1. Next we use the elementary transformation method and the inverse matrix method to give the other two solutions of example 1.
1 1 1 2 3 0 1 2 1 1 , . 1 0 1 2 2 1 3 2 0 5 AIt is then transformed into the simplest form of a row by an elementary transformation,
22 1 0 0 0 15 31 0 1 0 0 15 , 4 0 0 1 0 3 2 0 0 0 1 5 A .The row simplest matrix is transformed into a set of equations, and the solution is
1 2 3 4
22 31 4 2
, , , .
15 15 3 5
x x x x
Example 1 Method 3: First, write out the Coefficient Matrix A of the system of linear equations, the inverse matrix is obtained by applying the elementary transformation of the row.
Make the Matrix
1 1 1 2 1 0 0 0 0 1 2 1 0 1 0 0 , 1 0 1 2 0 0 1 0 1 3 2 0 0 0 0 1 A E .
Perform the elementary transformation on it, we have
3 11 4 1 15 15 5 3 7 8 1 2 15 15 5 3 1 2 1 3 3 3 1 1 1 5 5 5 1 0 0 0 0 1 0 0 , 0 0 1 0 0 0 0 0 1 0 A E
BecauseAbecomesE, which means 1
A exists, and 1
A is the matrix thatEbecomes.
To represent the system of linear equations
Ax
, we multiply left timesA
1 on both sides to get thesolution of (1), that is 1 22 31 4 2 , , , 15 15 3 5 T xA .
The above solutions to example 1 show us the role of the Matrix and its operations in the solution of the system of linear equations. We find that the elementary transformation of the rows of the Matrix is the simplest of the many methods for solving the system of linear equations; we can further summarize the relationship between the structure of the system of Linear Equations Solution and the rank of the Matrix.
2.3. Constructing the Knowledge thread of Vector Groups and System of Linear Equations
There are a lot of concepts involved in the Linear independence of a vector group; it is especially difficult for students to accept. Concepts like "linear combination" , "linear representation" , "vector group equivalence" , "linear correlation" , "linear independence", etc., there is a close relationship among vector group, Matrix and system of linear equations, so long as we pay attention to the integrity of the content and explain the relationship between them, we can effectively break through the focus and difficulty of the learning content [6].
Firstly, there is a one-to-one correspondence between a vector group and a Matrix; it is essentially a problem of partitioning a matrix by row or column. From the perspective of the Block Matrix, the Matrix of
n
rows andm
columns can be regarded as a vector group containingm
n
-dimensional column vectors. Conversely, a vector group containing a finite number of vectors can always form a matrix. The definition of this correspondence will pave the way for further study on the linear relation of vector group.
11 12 1 21 22 2 1 2 1 2 , , , m m m n n nm a a a a a a A a a a .According to the simplified form Ax of the system of linear equations, here's another way to represent system of linear equations:
1 2 1, 2, , m 1 1 2 2 m m . m x x x x x x By definition, a vector is a linear combination of a vector group A, x x1, 2,,xmare combination
coefficients, or the solution of a system of linear equations, the second group of corresponding relations can be obtained. A vectorcan be represented linearly by a vector group A, which is equivalent to having a system of linear equations solution. The fact that a vector cannot be represented linearly by a vector groupAmeans that the combination coefficient does not exist. Because of the relation between the rank of the Matrix and the solution of the system of linear equations, here’s an important theorem:
A vectorcan be represented linearly by a vector group
A
R A
R A
,
[7].On this basis, we further explore the linear relationship between vector groups. Two vector groups A: 1, 2,,mandB: 1, 2,,l , each vectorj,j1, 2,,l in the vector set B can be represented linearly by the vector set
A
,
11 12 1 21 22 2 1 2 1 2 1 2 , , , , , , l l l m m m ml x x x x x x x x x .Write it down as BAX, the existence of the combination coefficient means that the matrix equation AXB has a solution, It's Necessity and sufficiency is R A
R A B
,
.In the same way, each vector i,i1, 2,,m in the vector setAcan be represented linearly by the vector set
B, that isABK,the matrix equationBKA has a solution, and It's Necessity and sufficiency is R B
R B A, .So, two vector groups A: 1, 2,,mandB: 1, 2,,lare equivalent
R A
R B
R A B
,
.[4] We show it by an example.Example 2 Determine whether two vector groups
1 : 1, 1, 2 T, A 2
3, 0, 3
, T 3
1, 2, 0
Tand B:1
3, 5, 1
T, 2
1, 0, 6
T,3
1, 1, 3
T, 4
0, 1, 4
Tare equivalent or not.First, we make a Matrix
1 2 3 1 2 3 4 , , , , , , , ,
A B it is then transformed into a row-step Matrix by elementary transformation, as follows
1 3 1 3 1 1 0 , 0 3 3 8 1 2 1 0 0 7 17 7 1 1 A B .As a result,R A
R A B
,
3, and the MatrixB
has a third-order sub-formula1 1 0 0 1 1 0. 6 3 4
SoR B
3.Above all,R A
R B
R A B
,
3.Therefore the vector groupA B, is equivalent. Homogeneous system of linear equationsAx0is special circumstances of Ax , it must have a solution: only zero solutions and infinitely many solutions (non-zero solutions), these two cases correspond to the Linear independence of the vector group. From the definition, linear correlation shows that for a vector group A: 1, 2,,m , there are a series of constants k k1, 2,,km, which is not all zero, so that1 1 2 2 m m 0. k k k That means 1 2 1, 2, , m 0 m x x Ax x
has a non-zero solution, and it has
a nonzero solution
R A m, so the vector groupA: 1, 2,,m is linearly related
R A m[7]. Itsconverse proposition is that a vector group is linearly independent
R A m [7].In the course of Linear Algebra, system of linear equations and its related contents are the main lines of teaching. In matrices and their operations, Matrix multiplication provides a simple representation for system of linear equations. The elementary transformation of the Matrix gives the solution of the generalized system of Linear Equations, and summarizes the relation between the rank of the Matrix and the system of linear equations solution. The structure of the system of linear equations solution is discussed by the Linear independence of the vector group [8]. The similarity Matrix, quadratic form and linear transformation are the important knowledge points of the basic concepts and typical applications, in teaching, teachers should adopt heuristic and participatory teaching methods, grasp the main line of teaching from a macro perspective, and guide students from a micro perspective to comb the knowledge skeleton of each chapter, by revealing the relationship between different concepts and principles in a simple and easy way, the students gradually establish a clear knowledge structure in their minds, expand their thinking and improve their ability to analyze and deal with problems, effectively accomplish the mutual transformation of knowledge and problems, and enhance the sense of learning achievement.
III. C
ONCLUSION-rse of Linear Algebra, students are puzzled in their study, it also stems from the fact that it is difficult to find a situation in which linear Algebra can be used to solve practical problems. In the course of teaching, students combine the major they have studied with examples to illustrate the background and make them realize the usefulness of the course, helping students establish positive learning motivation is an important part of the teaching system, not just the inculcation of theories and methods [9]. In the course of Linear Algebra, the complicated operation is one of the reasons that students are afraid of it, and it will affect the efficiency of the class. Integrating Computer Technology and mathematical experiment into the course teaching can also stimulate students 'learning enthusiasm.
The relevance research of Linear Algebra course content is an effective way to solve its abstractness, and at the same time help students to deepen their grasp of basic knowledge. If, on the premise of analyzing the knowledge structure of the course, combining the course content, paying attention to the case study, breaking through the study difficult problem by means of the mathematical experiment, and achieving the harmony between teaching and learning, it can bring great impetus to the teaching practice of Linear Algebra and improve teaching quality and efficiency [10].
A
CKNOWLEDGMENTThe work is supported by Tai’an Science and Technology Development Project: Research on case teaching of linear algebra combined with learners practical ability (2019ZC251) and Teaching Reform and Research Project of Taishan University in 2018: The Theory of Probability Combining Real Life Application and Computer Simulation and the Teaching Reform of Mathematical Statistics Course.
R
EFERENCES[1] X.-Y. Meng, T. -F.Zhu. Relevance Research and practice of Linear Algebra course content. University education, 2019(07), pp. 118-120.
[2] C.-X.Wang. Research on the characteristics and teaching methods of Linear Algebra. University education, 2019 (11), pp. 91-93. [3] A.-J.Ren. Case Study of Linear Algebra teaching in Independent College. Mathematics learning and research, 2019 (21), pp.28. [4] Y.-S. Jin, R. Jia. Practice and exploration of teaching method reform of Linear Algebra. Journal of University of Science and
Technology Liaoning, 2013(5), pp. 541-544.
[5] L. Zhu, Q. -F. Jiang. A review of the teaching and research of Linear Algebra in foreign countries. Mathematics Education, 2018(1), pp.79-84.
[6] R.Jiang, S.-Z.Wang. The application of rank of Matrix in Linear Algebra and the discussion of its teaching method . Journal of Southwest Normal University (natural science edition) , 2012(8),pp.175-180.
[7] Department of Mathematics, Tongji University. Linear algebra. Fourth Edition. Beijing: Higher Education Press, 2003, pp.46 -110. [8] D. -K. Zhao. Some reflections on cultivating students 'mathematical thinking consciousness in linear algebra teaching. Advanced
Mathematics Research, 2017, 20(04), pp.110-112 + 116.
[9] H.-B. Zhao. Problem-driven is one of the effective teaching methods of Linear Algebra. Research in advanced mathematics, 2008(04), pp.91-94.
[10] S.-Z.Li. Introduced linear algebra concept from the problem. Advanced Mathematics Study, 2006(05), pp .6-8 + 15.
A
UTHOR’
SP
ROFILEFirst Author
Rui Chen is a lecturer at Taishan University, China. She obtained her master's degree from Shandong University in December, 2009. Her research interests are in the areas of application of probability theory, and applied statistics in recent years.
Second Author
Liang Fang was born in December 1970 in Feixian County, Linyi City, Shandong province, China. He is a professor at Taishan University. He obtained his PhD from Shanghai Jiaotong University in June, 2010. His research interests are in the areas of cone optimizations, numerical analysis, and complementarity problems.
