Linear Algebra, Theory And Applications, 2012a

56 MATRICES AND LINEAR TRANSFORMATIONS
where b is chosen to satisfy the equation
a 1 b +
n∑
a k =0
k=2
Suppose now that the theorem is true for any m × n matrix with n>mand consider an
(m × 1) × n matrix A where n>m+1. If the first column of A is 0, then you could let
x = e 1 as above. If the first column is not the zero vector, then by doing row operations,
the equation Ax = 0 can be reduced to the equivalent system
where A 1 is of the form
A 1 x = 0
(
1 a
T
A 1 =
0 B
where B is an m × (n − 1) matrix. Since n>m+1, it follows that (n − 1) >mand so
by induction, there exists a nonzero vector y ∈ F n−1 such that By = 0. Then consider the
vector
( ) b
x =
y
⎛ ⎞
b T 1
A 1 x has for its top entry the expression b + a T ⎜
y. Letting B = . ⎟
⎝
⎠ , the i th entry of
A 1 x for i>1 is of the form b T i y =0. Thusifb is chosen to satisfy the equation b+aT y =0,
then A 1 x = 0.
)
b T m
2.4 Subspaces And Spans
Definition 2.4.1 Let {x 1 , ··· , x p } be vectors in F n . A linear combination is any expression
of the form
p∑
c i x i
i=1
where the c i are scalars. The set of all linear combinations of these vectors is called
span (x 1 , ··· , x n ) . If V ⊆ F n , then V is called a subspace if whenever α, β are scalars
and u and v are vectors of V, it follows αu + βv ∈ V . That is, it is “closed under the
algebraic operations of vector addition and scalar multiplication”. A linear combination
of vectors is said to be trivial if all the scalars in the linear combination equal zero. A set
of vectors is said to be linearly independent if the only linear combination of these vectors
which equals the zero vector is the trivial linear combination. Thus {x 1 , ··· , x n } is called
linearly independent if whenever
p∑
c k x k = 0
k=1
it follows that all the scalars c k equal zero. A set of vectors, {x 1 , ··· , x p } , is called linearly
dependent if it is not linearly independent. Thus the set of vectors is linearly dependent if
there exist scalars c i ,i=1, ··· ,n, not all zero such that ∑ p
k=1 c kx k = 0.
Proposition 2.4.2 Let V ⊆ F n . Then V is a subspace if and only if it is a vector space
itself with respect to the same operations of scalar multiplication and vector addition.