Consider n linear equations
a11* x1+ a12 * x2 +…+ a1n * xn = b1
a21* x1+ a22 * x2 +…+ a2n * xn = b2
an1* x1+ an2 * x2 +…+ ann * xn = bn
There are several methods to asses and solve these equations . Finally by any method the solutions x1, x2 ,…, xn are linear combinations of b1, b2 ,…, bn . But
is it possible without any algebraic (or simpler) calculations to show that x1, x2 ,…, xn are such linear combinations?
I am thinking about concepts which predict solutions without a direct effort
to solve. That may be useful in some difficult situations ,of course no this simple example.