My lecturer gave me in my notes the proof that a system of linear equations cannot have two solutions but I don't understand it. Please give me yours with some explanations. Thanks.
-
A system of linear equations can have two solutions:
x+y = 0
2x + 2y = 0
has solutions (1, -1) and (2, -2). In fact it has infinitely many solutions, all of the form (a, -a).
If you mean that a system of linear equations cannot have *exactly* two solutions, that's actually true. Here's a proof [...over the reals...]:
Suppose that the linear system encoded in the matrix equation Ax = b has two distinct solutions, say u and v. I'll show that the system then has infinitely many solutions. Let c be any real number. The vector c*u + (1-c)*v is distinct for each c, so there are infinitely many such vectors. Moreover, each is a solution of the linear system:
A(cu + (1-c)v) = Acu + A(1-c)v
= c(Au) + (1-c)(Av)
= cb + (1-c)b
= cb + b - cb
= b
x+y = 0
2x + 2y = 0
has solutions (1, -1) and (2, -2). In fact it has infinitely many solutions, all of the form (a, -a).
If you mean that a system of linear equations cannot have *exactly* two solutions, that's actually true. Here's a proof [...over the reals...]:
Suppose that the linear system encoded in the matrix equation Ax = b has two distinct solutions, say u and v. I'll show that the system then has infinitely many solutions. Let c be any real number. The vector c*u + (1-c)*v is distinct for each c, so there are infinitely many such vectors. Moreover, each is a solution of the linear system:
A(cu + (1-c)v) = Acu + A(1-c)v
= c(Au) + (1-c)(Av)
= cb + (1-c)b
= cb + b - cb
= b