What is the Conceptual Understanding of Systems of Linear Equations ? What its the Difficulties During Solving Applied Economics, Physics, Engineering Problems? Recently I am working a paper in this field, but I can't find any studies for this area.
There are two primary approaches to visualise a linear system of equations:
1. Row wise picture
2. Column wise picture
Let the linear system be: A( m by n) *x( n by 1) = b(m by 1)
In the row wise visualization, we consider 'm' number of Hyperplanes of the form (ai'*x) = b, where a1( 1 by n),....., am(1 by n) are the rows of the matrix A( m by n). Each of these Hyperplanes is a convex set, more precisely a affine set, the linear system in this form of visualization is the set of all vectors x( n by 1), which satisfy each of these 'm' hyperplane equations simultaneously.... that is, they lie on the intersection of these Hyperplanes.... since we know that intersection of a finite number of convex sets is also a convex set... the set of all solutions x(n by 1) of the linear system Ax=b forms a convex set, which is a polyhedral set.
In the column wise visualization we observe that the right hand side vector b( m by 1) is a linear combination of the columns of the matrix A( m by n) :
A( m by n) = [ z1 z2 .... zn ] , zj ( m by 1) , j= 1,2,...,n are the columns of the matrix A
x ( n by 1) = ( x1 x2... xn) ' , where the symbol ' stands for transpose operation .
therefore A*x = z1*x1 + .... + zn*xn which is a vector of column space of A (m by n) for all x1, x2,..., xn
Thus, we can see that the linear system Ax=b has a solution if and only if the vector b ( m by 1) lies in the column space of the matrix A.
If it does, and the columns of the matrix A are forming a linearly independent set, that is, z1,...., zn is a basis for the Column space of matrix A, then, the solution vector x( n by 1) is unique. Otherwise, multiple x vector can satisfy the linear system, which have a unique component w( n by 1) belonging to the row space of the matrix A, and any vector y ( n by 1) belonging to the null space of the matrix A.
x ( n by 1) = w( n by 1) + y( n by 1)
w(n by 1) = G( n by m) *b ( m by 1),
where G( n by m) is the pseudo inverse of the matrix A ( m by n)
for those linear systems where solution does not exist, one can find an approximate solution xbar ( n by 1) by minimising the residual vector
r= b - (A*x) , as a function of 'x', in a pre-chosen norm function...... solving the above problem in the 2-norm is called the "Least squares solutions" to the minimisation problem.
Other than theoretical, there are several computational issues associated with solving the systems of linear equations.... issues with numerical sensitivity to small perturbations associated with computations.... these issues are addressed through numerical measures such as the condition number of the matrix A and other such measures.
An error in the writing above:
The Hyperplane equations are: ai( 1 by n) *x (n by 1) = bi
where i= 1,2,...,m
b( m by 1) = ( b1 b2... bm) '
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