Definniton. Suppose that V is a vector space. This
means it could be a
function space or a subset of R^n that is closed under addition and scalar
multiplication . A basis for V is a spanning set of minimal size.
It turns out, that for any vector space the number of
elements in any two
basis is always the same. We will prove this in a bit. This number is called
the dimension of the vector space.
We have seen that if you solve a homogeneous linear ODE ,
then the set
of solutions is a function space, and if you solve a homogeneous system of
linear equations , the set of solutions is a vector space. When we solve
nonhomogeneous
linear ODEs or nonhomogeneous systems of linear equations,
the set of solutions is obtained from one particular solution by adding all the
solutions in the vector space of the associated homogeneous equations.
1. (a) Find a basis for the fol lowing homogeneous system
of linear equations
x + 2y + 3z = 0
x + y + z = 0
3x + 5y + 7z = 0
(b) Use your basis to find all solutions to
x + 2y + 3z = 6
x + y + z = 3
3x + 5y + 7z = 15
(c) Use your basis to find all solutions to
x + 2y + 3z = 6
x + y + z = 3
3x + 5y + 7z = 10
(d) Find a basis for the set of solutions to thes
following homogenous system
(e) Use your basis to find all solutions to
where 
and
are arbitrary constants.
(f) Compare what happened in (e) to what happened in (b)
and (c).
2. Think about what happened in 1 (d) above.
(a) Find a system of equations whose soluton is the span
of the set of vectors:
(b) Find a system of equations whose soluton is the span
the of set of vectors:
3.
(a) Show that a mininmal spanning set of a vector space consists of linearly
independent vectors.
(b) Take a set of vectors, such as those in problem 2. Use them as the rows
of a matrix. Show that “ row operations ” don’t change the span of the rows.
(c) Show that the non zero rows of a reduced echelon matrix are linearly
independent.
(d) For any system of homogeneous linear equations explain why there is a
unique reduced row-echelon matrix R such that the set of solutions to
is the same as that of the original system. (Hint: compare the homogeneous
solution sets to two different homogeneous reduced row-echelon systems.)
