西安建筑科技大学：《线性代数》课程双语教学课件_线性相关与线性无关

The Structure of solutions First we look at a homogeneous system a111+a12X+…+a1nXn=0 a21X1+a22X+…+a2nxn=0 am1x1 t am2X2 +..+ amn=0

The Structure of Solutions First we look at a homogeneous system    a11x1 + a12x2 + · · · + a1nxn = 0 a21x1 + a22x2 + · · · + a2nxn = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . am1x1 + am2x2 + · · · + amnxn = 0 (2) The set of solutions of a system of linear equations is called the solution set. The solution set of a homogenous system is also called the solution space. If it has zero solution only, the structure is simple and we are not interested in this case. So we suppose that the rank r of the coefficient matrix of (2) is less than n. Then the system has infinitely many solutions. () April 28, 2006 1 / 15

The Structure of solutions First we look at a homogeneous system a111+a12X+…+a1nXn=0 a21X1+a22X+…+a2nxn=0 am1X1 t am2X2+..+ amnon=0 o The set of solutions of a system of linear equations is called the solution set

The Structure of Solutions First we look at a homogeneous system    a11x1 + a12x2 + · · · + a1nxn = 0 a21x1 + a22x2 + · · · + a2nxn = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . am1x1 + am2x2 + · · · + amnxn = 0 (2) The set of solutions of a system of linear equations is called the solution set. The solution set of a homogenous system is also called the solution space. If it has zero solution only, the structure is simple and we are not interested in this case. So we suppose that the rank r of the coefficient matrix of (2) is less than n. Then the system has infinitely many solutions. () April 28, 2006 1 / 15

The Structure of solutions First we look at a homogeneous system a111+a12X+…+a1nXn=0 a21X1+a22X+…+a2nxn=0 am1X1 t am2X2+..+ amnon=0 o The set of solutions of a system of linear equations is called the solution set o The solution set of a homogenous system is also called the solution space

The Structure of Solutions First we look at a homogeneous system    a11x1 + a12x2 + · · · + a1nxn = 0 a21x1 + a22x2 + · · · + a2nxn = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . am1x1 + am2x2 + · · · + amnxn = 0 (2) The set of solutions of a system of linear equations is called the solution set. The solution set of a homogenous system is also called the solution space. If it has zero solution only, the structure is simple and we are not interested in this case. So we suppose that the rank r of the coefficient matrix of (2) is less than n. Then the system has infinitely many solutions. () April 28, 2006 1 / 15

The Structure of solutions First we look at a homogeneous system a111+a12X+…+a1nXn=0 a21X1+a22X+…+a2nxn=0 am1X1 t am2X2+..+ amnon=0 o The set of solutions of a system of linear equations is called the solution set o The solution set of a homogenous system is also called the solution space o If it has zero solution only, the structure is simple and we are not interested in this case

The Structure of Solutions First we look at a homogeneous system    a11x1 + a12x2 + · · · + a1nxn = 0 a21x1 + a22x2 + · · · + a2nxn = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . am1x1 + am2x2 + · · · + amnxn = 0 (2) The set of solutions of a system of linear equations is called the solution set. The solution set of a homogenous system is also called the solution space. If it has zero solution only, the structure is simple and we are not interested in this case. So we suppose that the rank r of the coefficient matrix of (2) is less than n. Then the system has infinitely many solutions. () April 28, 2006 1 / 15

The Structure of solutions First we look at a homogeneous system a111+a12X+…+a1nXn=0 a21X1+a22X+…+a2nxn=0 am1X1 t am2X2+..+ amnon=0 o The set of solutions of a system of linear equations is called the solution set o The solution set of a homogenous system is also called the solution space o If it has zero solution only, the structure is simple and we are not interested in this case o So we suppose that the rank r of the coefficient matrix of(2 )is less than n Then the system has infinitely many solutions

The Structure of Solutions First we look at a homogeneous system    a11x1 + a12x2 + · · · + a1nxn = 0 a21x1 + a22x2 + · · · + a2nxn = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . am1x1 + am2x2 + · · · + amnxn = 0 (2) The set of solutions of a system of linear equations is called the solution set. The solution set of a homogenous system is also called the solution space. If it has zero solution only, the structure is simple and we are not interested in this case. So we suppose that the rank r of the coefficient matrix of (2) is less than n. Then the system has infinitely many solutions. () April 28, 2006 1 / 15

The Structure of solutions A solution(c1, c2, .. cn)of(2 )is a vector of dimension n. It is also called a solution vector of (2) First we note two basic facts

The Structure of Solutions A solution (c1, c2, · · · , cn) of (2) is a vector of dimension n. It is also called a solution vector of (2). First we note two basic facts: If γ = (c1, c2, · · · , cn) is a solution of (2), then kγ = k(c1, c2, · · · , cn), where k is any number in the field F, is also a solution of (2). If γ1 = (c11, · · · , c1n) and γ2 = (c21, · · · , c2n) are solutions of (2), then γ1 + γ2 = (c11 + c21, · · · , c1n + c2n) is also a solution of (2). () April 28, 2006 2 / 15

The Structure of solutions A solution(c1, c2,..., cn)of(2) is a vector of dimension n. It is also called a solution vector of (2) First we note two basic facts o If ?=(c1, c2,.Cn)is a solution of (2), then ky=k(c1, C2, .. cn), where k is any number in the field F, is also a solution of (2 )

The Structure of Solutions A solution (c1, c2, · · · , cn) of (2) is a vector of dimension n. It is also called a solution vector of (2). First we note two basic facts: If γ = (c1, c2, · · · , cn) is a solution of (2), then kγ = k(c1, c2, · · · , cn), where k is any number in the field F, is also a solution of (2). If γ1 = (c11, · · · , c1n) and γ2 = (c21, · · · , c2n) are solutions of (2), then γ1 + γ2 = (c11 + c21, · · · , c1n + c2n) is also a solution of (2). () April 28, 2006 2 / 15

The Structure of solutions A solution(c1, c2,..., cn)of(2) is a vector of dimension n. It is also called a solution vector of (2) First we note two basic facts o If y=(c1, c2,..., cn)is a solution of(2), then ky=k(c1, c2, .. cn), where k is any number in the field F, is also a solution of (2 ). If =(c1l,., Cin)and 72=(c21,.., c2n )are solutions of(2), then is also a solution

The Structure of Solutions A solution (c1, c2, · · · , cn) of (2) is a vector of dimension n. It is also called a solution vector of (2). First we note two basic facts: If γ = (c1, c2, · · · , cn) is a solution of (2), then kγ = k(c1, c2, · · · , cn), where k is any number in the field F, is also a solution of (2). If γ1 = (c11, · · · , c1n) and γ2 = (c21, · · · , c2n) are solutions of (2), then γ1 + γ2 = (c11 + c21, · · · , c1n + c2n) is also a solution of (2). () April 28, 2006 2 / 15

The Structure of solutions Let s be the set of all solutions of (2). The set s has rank r, where s <n Therefore there are s vectors1,……,s∈ s forming a maximal linearly independent subset of S It follows that every solution of (2)is a linear combination of all solutions of (2)is given by k1m1+k272+…+ks where k,,..., ks are arbitrary numbers in the field f. We call the set of these s solutions a basis for solutions of the system

The Structure of Solutions Let S be the set of all solutions of (2). The set S has rank r, where s ≤ n. Therefore there are s vectors γ1, · · · , γs ∈ S forming a maximal linearly independent subset of S. It follows that every solution of (2) is a linear combination of γ1, · · · , γs . So all solutions of (2) is given by k1γ1 + k2γ2 + · · · + ksγs where k1, · · · , ks are arbitrary numbers in the field F. We call the set of these s solutions γ1, · · · , γs a basis for solutions of the system. () April 28, 2006 3 / 15

The Structure of solutions We need only to find a basis of solutions for solving a syste homogeneous linear equations although the system has infinitely many solutions What can we say about The following theorem answers this question Theorem The rank of the solutions equals n-r, where n is the number of indeterminate and r is the rank of the coefficient matrix

The Structure of Solutions We need only to find a basis of solutions for solving a system of homogeneous linear equations although the system has infinitely many solutions. What can we say about ? The following theorem answers this question. Theorem The rank of the solutions equals n − r, where n is the number of indeterminate and r is the rank of the coefficient matrix. () April 28, 2006 4 / 15