2
1 Some Linear Algebra
(2) The operations · and + satisfy
(2.1) ∀ a ∈ K , ∀ v, w ∈ V , a(v + w) = av + aw;
(2.2) ∀ a, b ∈ K and ∀ v ∈ V , (a + b)v = av + bv;
(2.3) ∀ a, b ∈ K and ∀ v ∈ V , (ab)v = a(bv);
(2.4) ∀ v ∈ V , 1v = v.
Remark 1.1.1 (1) In order to avoid stress of notation we will denote only by V the
vector space (V, +, ·) in the cases where there is no possibility of confusion.
(2) Throughout this chapter we always assume that all vector spaces are defined
over the same field K .
Subspaces of a vector space V are subsets of V which are also vector spaces under
the operations of V .
Definition 1.1.2 Let V be a vector space. A subset W ⊆ V is a subspace of V if the
following conditions hold:
(1) 0 ∈ W ;
(2) W is closed under vector addition in V , i.e., ∀ v, w ∈ W , it follows that v + w ∈
W ;
(3) W is closed under scalar multiplication, i.e., ∀ v ∈ W and ∀ a ∈ K , it implies
that av ∈ W .
Definition 1.1.3 Let V be a vector space and consider that v, v 1 , v 2 , . . . v n ∈ V . We
say that v is a linear combination of the vectors v 1 , v 2 , . . . , v n if v can be written as
v =
n
i=1
a i v i , where a i ∈ K , for i = 1, 2, . . . , n.
Definition 1.1.4 Let V be a vector space and let S = {v 1 , v 2 , . . . , v n } be a set of
vectors in V . Then the set v 1 , v 2 , . . . , v n is the set of all linear combinations of
v 1 , v 2 , . . . , v n , called subspace spanned by S.
The following result characterizes subspaces spanned by a set.
Theorem 1.1.1 Let V be a vector space and S = {v 1 , v 2 , . . . , v n } be a set of vectors
in V . Then the intersection of all the subspaces of V containing S is v 1 , v 2 , . . . , v n ,
i.e.,
v 1 , v 2 , . . . , v n =
S⊆W ⊆V
W,
where W runs through the subspaces of V containing S. Hence, v 1 , v 2 , . . . , v n is
the smallest subspace of V which contains S.
Exercise 1.1.1 Show Theorem 1.1.1.
Definition 1.1.5 Let V be a vector space and v 1 , v 2 , . . . , v n ∈ V . We say that the set
S = {v 1 , v 2 , . . . , v n } is linearly independent (LI) (or that the vectors v 1 , v 2 , . . . , v n
1 Some Linear Algebra
(2) The operations · and + satisfy
(2.1) ∀ a ∈ K , ∀ v, w ∈ V , a(v + w) = av + aw;
(2.2) ∀ a, b ∈ K and ∀ v ∈ V , (a + b)v = av + bv;
(2.3) ∀ a, b ∈ K and ∀ v ∈ V , (ab)v = a(bv);
(2.4) ∀ v ∈ V , 1v = v.
Remark 1.1.1 (1) In order to avoid stress of notation we will denote only by V the
vector space (V, +, ·) in the cases where there is no possibility of confusion.
(2) Throughout this chapter we always assume that all vector spaces are defined
over the same field K .
Subspaces of a vector space V are subsets of V which are also vector spaces under
the operations of V .
Definition 1.1.2 Let V be a vector space. A subset W ⊆ V is a subspace of V if the
following conditions hold:
(1) 0 ∈ W ;
(2) W is closed under vector addition in V , i.e., ∀ v, w ∈ W , it follows that v + w ∈
W ;
(3) W is closed under scalar multiplication, i.e., ∀ v ∈ W and ∀ a ∈ K , it implies
that av ∈ W .
Definition 1.1.3 Let V be a vector space and consider that v, v 1 , v 2 , . . . v n ∈ V . We
say that v is a linear combination of the vectors v 1 , v 2 , . . . , v n if v can be written as
v =
n
i=1
a i v i , where a i ∈ K , for i = 1, 2, . . . , n.
Definition 1.1.4 Let V be a vector space and let S = {v 1 , v 2 , . . . , v n } be a set of
vectors in V . Then the set v 1 , v 2 , . . . , v n is the set of all linear combinations of
v 1 , v 2 , . . . , v n , called subspace spanned by S.
The following result characterizes subspaces spanned by a set.
Theorem 1.1.1 Let V be a vector space and S = {v 1 , v 2 , . . . , v n } be a set of vectors
in V . Then the intersection of all the subspaces of V containing S is v 1 , v 2 , . . . , v n ,
i.e.,
v 1 , v 2 , . . . , v n =
S⊆W ⊆V
W,
where W runs through the subspaces of V containing S. Hence, v 1 , v 2 , . . . , v n is
the smallest subspace of V which contains S.
Exercise 1.1.1 Show Theorem 1.1.1.
Definition 1.1.5 Let V be a vector space and v 1 , v 2 , . . . , v n ∈ V . We say that the set
S = {v 1 , v 2 , . . . , v n } is linearly independent (LI) (or that the vectors v 1 , v 2 , . . . , v n