A system of m linear equations in n variables is equivalent to a single matrix equation of the form
A x = b ,
where A is an m \times n matrix, x is a column of length n, and b is a column of length m.The expression A x indicates the result of the ``matrix multiplication'' of A and x.
For a given m \times n matrix A and any column x of length n, the expression
L_{A}(x) = A x
defines a function L_{A}.The function L_{A} is a linear function.
Definition. A function f in this context is linear if it obeys the two rules:
f(x1 + x2) = f(x1) + f(x2) for all x_{1}, x_{2}.
f(u x) = u f(x) for all numbers u and all points x.
The exercises below may be done by using common sense and secondary school technique.
Find all solutions of the system of equations
| x - 2 y + z = 0 |
| 5 x - 4 y + 3 z = 0 |
| 3 x - 3 y + 2 z = 0 |
The exercises below pertain to the following system of equations:
| x - 2 y + z = a |
| 5 x - 4 y + 3 z = b |
| 3 x - 3 y + 2 z = c |
Find a non-zero point (a, b, c) for which the system has no solution.
Find a non-zero point (a, b, c) for which the system has at least one solution.
Describe geometrically the set of all points (a, b, c) for which the system has at least one solution.