Page 61 - Linear Algebra Done Right

Null Spaces and Ranges
n
n

a 1,k x k ,...,
a m,k x k .
T(x 1 ,...,x n ) =
k=1
k=1
Now consider the equation Tx = 0 (where x ∈ F n and the 0 here is 47
m
the additive identity in F , namely, the list of length m consisting of
all 0’s). Letting x = (x 1 ,...,x n ), we can rewrite the equation Tx = 0
as a system of homogeneous equations: Homogeneous, in this
n context, means that the

a 1,k x k = 0 constant term on the
k=1 right side of each
. . . equation equals 0.
n

a m,k x k = 0.
k=1
We think of the a’s as known; we are interested in solutions for the
variables x 1 ,...,x n . Thus we have m equations and n variables. Obvi-
ously x 1 =· · ·= x n = 0 is a solution; the key question here is whether
any other solutions exist. In other words, we want to know if null T is
strictly bigger than {0}. This happens precisely when T is not injective
(by 3.2). From 3.5 we see that T is not injective if n>m. Conclusion:
a homogeneous system of linear equations in which there are more
variables than equations must have nonzero solutions.
With T as in the previous paragraph, now consider the equation
m
Tx = c, where c = (c 1 ,...,c m ) ∈ F . We can rewrite the equation
Tx = c as a system of inhomogeneous equations:
n

a 1,k x k = c 1
These results about
k=1 homogeneous systems
. . . with more variables
n than equations and

a m,k x k = c m . inhomogeneous
k=1 systems with more
As before, we think of the a’s as known. The key question here is equations than
whether for every choice of the constant terms c 1 ,...,c m ∈ F, there variables are often
exists at least one solution for the variables x 1 ,...,x n . In other words, proved using Gaussian
m
we want to know whether range T equals F . From 3.6 we see that T elimination. The
is not surjective if n<m. Conclusion: an inhomogeneous system of abstract approach
linear equations in which there are more equations than variables has taken here leads to
no solution for some choice of the constant terms. cleaner proofs.