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.
At first I apologize all forgetting to notify that existence and uniqueness of answers has been supposed .Consider other equations with answers x'1, x'2 ,…, x'n as
following :
a11* x'1+ a12 * x'2 +…+ a1n * x'n = b'1
a21* x'1+ a22 * x'2 +…+ a2n * x'n = b'2
.
.
.
an1* x'1+ an2 * x'2 +…+ ann * x'n = b'n
Then regard the following equations :
a11* x"1+ a12 * x"2 +…+ a1n * x"n = C* b1 + D * b'1
a21* x"1+ a22 * x"'2 +…+ a2n * x"n = C* b2 + D * b'2
.
.
.
an1* x"1+ an2 * x"2 +…+ ann * x"n = C* bn + D * b'n
Uniqueness of answers shows that :
x"i = C* xi + D * x'i , i =1 ,2,…,n
We can regard b1,b2,…, bn as a function on X ={1,2,…,n} such that :
bi = f(i) , i=1,2,…,n f: X ---- > C (complex values)
X along its all subsets make a locally compact Hausdorff topological space. All functions X ---- > C are continuous by this topology . For each function such as f,there are x1,x2,….,xn that satisfy the linear equations . If we regard one of them ,e.g. xm ,xm is a linear functional of f. This functional is bounded and is defined on Cc(X).So according to Friesz theorem (real and complex analysis Walter Rudin , third edition ,19.6) ,there is a measure such as M (along some properties)that xm= int (fdM)on X . Clearly this integral can regard as a linear combination of b1,b2,…, bn .