Fundamentals of Matrix Algebra, 2011a

2.4 Vector Soluons to Linear Systems
y
⃗v
.
x
⃗x p
.
.
Figure 2.20: The soluon, as a line, to A⃗x = ⃗b in Example 46.
The previous examples illustrate some important concepts. One is that we can
“see” the soluon to a system of linear equaons in a new way. Before, when we had
infinite soluons, we knew we could arbitrarily pick values for our free variables and
get different soluons. We knew this to be true, and we even pracced it, but the
result was not very “tangible.” Now, we can view our soluon as a vector; by picking
different values for our free variables, we see this as mulplying certain important
vectors by a scalar which gives a different soluon.
Another important concept that these examples demonstrate comes from the fact
that Examples 44 and 46 were only “slightly different” and hence had only “slightly
different” answers. Both soluons had
x 2
[ −2
1
in them; in Example 46 the soluon also had another vector added to this. Was this
coincidence, or is there a definite paern here?
Of course there is a paern! Now . . . what exactly is it? First, we define a term.
]
.
Ḋefinion 17
Homogeneous Linear System of Equaons
.
A system of linear equaons is homogeneous if the constants
in each equaon are zero.
Note: a homogeneous system of equaons can be wrien
in vector form as A⃗x = ⃗0.
The term homogeneous comes from two Greek words; homo meaning “same” and
genus meaning “type.” A homogeneous system of equaons is a system in which each
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