Basic Terminology for Systems of Equations in a Nutshell
A system of linear equations is something like the following:
3x1 − 7x2+4 x 310
5x +8 x − 12x = −1:
1 2 3
Note that the number of equations is not required to be the same as the number of unknowns.
A solution to this system would be a set of values for x x ,and x which makes the equations
1 2 3
true. For instance, 1 =3, x 21, x 3 2 is a solution. We will often think of a solution as being a
vector: [3; 1; 2] is a solution to the above equation. (For technical reasons, it will later be better to
write solution vectors vertically rather than horizontally. For the moment, we won’t worry about the
way vectors are written.)
As you know from Math 231, a system of two equations can also be thought of as a single
equation between two-dimensional vectors. It’s easier to see things if we write these vectors vertically.
The above system then becomes the vector equation
▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯
3 −7 4 10
5 x1 + 8 x 2+ −12 x3 = −1 :
A system of linear equations is called homogeneous if the right hand side is the zero vector. For
3x1 − 7x2+4 x 30
5x1 +8 x2− 12x 30 :
2 3 2 3
This system actually has a number of solutions, but there is one obvious one, namely2 5 = 4 05 :
This solution is called the trivial solution.( Important Note: Trivial as used this way in Linear
Algebra is a technical term which you need to know.)
De▯nition. A vector is called trivial if all its coordinates are 0, i.e. if it is the zero vector.
In Linear Algebra we are not interested in only ▯nding one solution to a system of linear
equations. We are interested in all possible solutions. In particular, homogeneous systems of
equations (see above) are very important. The important question for a homogeneous system is
whetherornotthereisany non-trivial solution, i.e. whether there is any solution other than the
trivial one. Sometimes there will be, sometimes there won’t. Paradoxically, it’s actually the case
where the trivial solution is the only possible one that is the most important. This situation is
described by one of the most important words in the whole course.
De▯nition. Let v 1;:::; n be vectors (all the same dimension). These vectors are called linearly
independent if the vector equation
x1v1+ ▯▯▯ + xnvn= 0
has only the trivial solution. 2
▯ ▯ ▯ ▯
Example. The vectors 1 and 3 are linearly independent.
How can we prove that the above is correct? We look for solutions to the linear system
▯ ▯ ▯ ▯ ▯ ▯
x 1 + x 3 = 0 ;
1 1 2 2 0
which can also be written as
x1+3 x 20
x1 +2 x2=0 :
Subtracting the ▯rst equation from the second shows th2t x = 0 and substitution then shows that
x1= 0. Therefore the only solution is the trivial one, so the vectors are linearly independent.
▯ ▯ ▯ ▯
Example. The vectors 2 and 6 are not linearly independent.
When we try high school algebra on the system
x1+3 x 20
2x1 +6 x2=0
it doesn’t work. We are unable to ▯nd a single unique solution. In fact, alt1ough x2=0 is
one solution, there is also another, non-trivial solu1ion x x2= −1. This non-trivial solution
shows that the vectors are not linearly independent.
Going back to non-homogeneous systems. (I.e. the right hand side is not zero.) Unlike
homogeneous systems, a non-homogeneous system might not have any solution at all. For instance,
the non-homogeneous system
2 3 2 3 2 3 2 3
1 3 −5 0
4 1 x1 + 4−2 5 x2 + 4 4 5 x3 = 4 2
2 1 −1 0
has no solution, since on the left-hand side the third coordinate of each vector is the sum of the ▯rst
two, but on the right hand side this is not true, 1o 2f x , x3satisfy the ▯rst two equations,
they cannot satisfy the third. We then say that this system of equations is inconsistent.
De▯nition. Let v 1;:::; n and b be vectors of the same size, written vertically. The system of
linear equations1x 1 ▯▯▯ + xnv n b is called consistent if it has at least one solution and is
called inconsistent if it has no solution.
If the system 1 1 + ▯▯▯ + n n = b is consistent, we say that b is a linear combination of
v1;:::; n . 3
Example. The linear system
▯ ▯ ▯ ▯ ▯ ▯
2 3 8
x1 + x 2 =
5 −1 3
▯ ▯ ▯ ▯
is consistent since it has the solutio=1 ;x = 2. Therefore 8 is a linear combination of 2
1 2 3 5
and 3 .
For ahomogeneous system of linear equations either
(1) the system has only one solution, the trivial one;
(2) the system has more than one solution.
For anon-homogeneous system either
(1) the syste