Math Forum / Mathematics / Undergraduate Math / March 2005

Not dependent.  Has no solution.  One multiple of other.

One or none.

Usually as many as the whole space.

1 Let the equations be

    ax + by = h
    cx + dy = k.

 If the vectors (a, b) and (c, d) are linearly independent, the
 equations are said to be independent.  (a, b) and (c, d) are said to
 be linearly independent if for no scalars p and q does

    p(a, b) + q(c, d) = 0

 hold.  Geometrically, this means that

    ax + by = h
    cx + dy = k

 are two non-parallel lines.

 The equations are inconsistent if they have no solution.  
 Geometrically, this means that

    ax + by = h
    cx + dy = k

 are two parallel lines.

 The equations are dependent if they are not independent.

2 If there are m linearly independent homogeneous linear equations
 in n unknowns, then there are n - m linearly independent solutions.

 If there are m non-homogeneous linear equations in m unknowns then
 there is one solution iff the matrix of the coefficients (i.e.

    [a b]
    [c d]

 in the case above) is non-singular.

3 I don't know.  What's a region?

Note that question 1 says "What does it mean to _you_ ...?"  Only you
can answer that.  If you reply "Absolutely nothing." and that is the
correct answer, then you should get full marks.