Linear Algebra

22 Chapter 1. Linear Systems
Studying the associated homogeneous system has a great advantage over
studying the original system. Nonhomogeneous systems can be inconsistent.
But a homogeneous system must be consistent since there is always at least one
solution, the vector of zeros.
3.4 Definition A column or row vector of all zeros is a zero vector, denoted �0.
There are many different zero vectors, e.g., the one-tall zero vector, the two-tall
zero vector, etc. Nonetheless, people often refer to “the” zero vector, expecting
that the size of the one being discussed will be clear from the context.
3.5 Example Some homogeneous systems have the zero vector as their only
solution.
3x +2y + z =0
6x +4y =0
y + z =0
−2ρ1+ρ2
−→
3x +2y + z =0
−2z =0
y + z =0
ρ2↔ρ3
−→
3x +2y + z =0
y + z =0
−2z =0
3.6 Example Some homogeneous systems have many solutions. One example
is the Chemistry problem from the first page of this book.
The solution set:
7x − 7j =0
8x + y − 5j − 2k =0
y − 3j =0
3y − 6j − k =0
−(8/7)ρ1+ρ2
−→
−ρ2+ρ3
−→
−3ρ2+ρ4
−(5/2)ρ3+ρ4
−→
⎛ ⎞
1/3
⎜
{ ⎜ 1 ⎟
⎝1/3⎠
1
w � � k ∈ R}
7x − 7z =0
y +3z − 2w =0
y − 3z =0
3y − 6z − w =0
7x − 7z =0
y + 3z − 2w =0
−6z +2w =0
−15z +5w =0
7x − 7z =0
y + 3z − 2w =0
−6z +2w =0
0=0
has many vectors besides the zero vector (if we interpret w as a number of
molecules then solutions make sense only when w is a nonnegative multiple of
3).
We now have the terminology to prove the two parts of Theorem 3.1. The
first lemma deals with unrestricted combinations.