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