Operator Algebras
نویسنده
چکیده
Notice that the left-hand side of the third equation is the sum of the left-hand sides of the first two. As a result, no solution to the system exists unless a + b = c. But if a + b = c, then any solution of the first two equations is also a solution of the third; and in any linear system involving more unknowns than equations, solutions, when they exist, are never unique. In the present case, if (x, y, z) is a solution, then so is (x + t, y + t, z + t), for any t. Thus the same phenomenon (a linear relation among the equations) which prevents the system from admitting solutions in some cases, also prevents solutions from being unique in other cases. To make the relation between existence and uniqueness of solutions more precise, consider a general system of linear equations of the form
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