Deﬁnition: The linear system Ax = 0 is said to be homoge-
neous.
The solution x = 0 is called the trivial solution.
Nonzero solutions are called nontrivial solutions.
Theorem: If 1 ,v2, ··· ,vkare solutions of the homogeneous linear
Xk
system Ax = 0, the linear combination λiv is also a solution
i=1
for any scalars 1 ,λ2, ··· ,λk.
Deﬁnition: A nonempty subset W of R is called a subspace of
R if the following conditions hold.
(i) For all u, w ∈ W, u + w ∈ W.
(closed under vector addition)
(ii) For all w ∈ W and λ ∈ R, λw ∈ W.
(closed under scalar multiplication) Corollary: Let W be the solution set of a homogeneous linear sys-
n n
tem Ax = 0, where x ∈ R . Then W is a subspace of R .
Deﬁnition: The subspace, W = { 0 } of R is called the zero
subspace.
n n
Deﬁnition: All subspaces of R , other than R itself, are called
n
proper subspaces of R .
Theorem: If W = sp(w ,w , 1·· 2w ) is thekspan of k vectors in
R , then W is a subspace of R . n
Deﬁnition: We say that the vectors w ,w ,1·· 2w , as ik the the-
orem, generate or span the subspace W = sp(w ,w , ··· ,w ).
1 2 k
a a ··· a
11 12 1k
21 a22 ··· a2k
Let A = ∈ M (R).
n,k
.
an1 an2 ··· ank
The span of the row vectors of A,
sp([a ,a , ··· ,a ], ··· ,[a ,a ,··· ,a ]),
11 12 1k n1 n2 nk
is called the row space of A. It is a subspace of R .
The span of theolum vector of
a11 a1k
a21 a2k
sp ··· ,
. .
. .
an1 ank
is called the column space of A. It is a subspace of R .
The solution set of Ax = 0 is called the nullspace of A. It is a
subspace of R . Remark: A linear system Ax = b has a solution if and only if b is
in the column space of A.

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