System of linear equation with no solution.

Question

I am trying to give reason why a system of linear equation may not have solutions in terms of vectorspaces and subspaces. Can any one give hint in this direction

Answer 1

Parallel lines, parallel planes.

Answer 2

Hint: consider dimensions of a vector space that is spanned on columns of matrix $A$ (where $Ax=b$) and a vector space that is spanned on columns of matrix $A$ and vector $b$

Answer 3

I would like to add onto Martins answer that no solution could mean parallel lines and parallel planes, but also include "skew lines", that is, lines which are not parallel but do not intersect. These skew lines may only exist in three or more dimensions. For example (courtesy of Wolfram MathWorld):

Answer 4

Consider the span of the column vectors of a matrix $A$. Now the link between the (geometric) notion of a vector- or subspace and linear equations, that is calculating, is to interpret the multiplication of $A$ with a vector $\vec v$ as a linear combination of the column vector of $A$, that is an element of the mentioned span.

Now, given a vector $\vec b$, that vector may be member of the span or not (that's the geometric side): if it is a member, it is a linear combination of the column vectors of $A$, that means exactly that $A\vec v=\vec b$ has a solution, otherwise not. And that's the calculation's side.

Answer 5

The general result is this:

In a finite dimensional vector space, a linear system $A\mathbf x=\mathbf b$ has solutions if and only if the matrix $A$ and the augmented matrix $A\mathbf b$ have the same rank.

Furthermore, if it is the case, the set $S$ of solutions is an affine subspace, and the common rank of $A$ and the augmented matrix is the codimension of $S$.