1.1 Introduction
3
are linearly independent) if the equation
n
i=1
a i v i = 0 implies that a i = 0 for all
i = 1, 2, . . . , n. If there exists at least one a i = 0 such that
n
i=1
a i v i = 0, then S (or
that the vectors v 1 , v 2 , . . . , v n ) is called linearly dependent (LD).
Definition 1.1.6 Let V be a vector space. A basis for V is a linearly independent set
of vectors in V which spans V . We say that V is finite-dimensional if it has a finite
basis.
Theorem 1.1.2 (Invariance of dimension) Let V be a vector space with bases S =
{v 1 , v 2 , . . . , v n } and T = {w 1 , w 2 , . . . , w m }. Then it follows that m = n.
Because of the invariance of dimension (Theorem 1.1.2), one can define the dimension of a vector space.
Definition 1.1.7 The dimension dim K (V ) of a finite-dimensional vector space V
(over K ) is the number of elements in a basis of V .
1.2 Linear Transformations
Here, we recall the concepts of linear transformation and linear operator.
Definition 1.2.1 Let V and W be two vectors spaces over the same field K . A
function T : V −→ W is a linear transformation from V into W if the following
hold:
(T1) ∀ u, v ∈ V , T (u + v) = T (u) + T (v);
(T2) ∀ v ∈ V , and ∀ a ∈ K , T (av) = aT (v).
In Linear Algebra, isomorphic vector spaces are essentially the same.
Definition 1.2.2 Let V and W be two vectors spaces. We say that V and W
are isomorphic, written V ∼ = W , if there exists a bijective linear transformation
T : V −→ W .
Exercise 1.2.1 Show that if V and W are finite-dimensional vector spaces then
V ∼ = W if and only if dim K (V ) = dim K (W ).
Theorem 1.2.1 Let V and W be vector spaces and let v 1 , v 2 , . . . , v n be a basis
of V . Given any list of vectors w 1 , w 2 , . . . , w n in W , there exists a unique linear
transformation T : V −→ W such that T (v i ) = w i for all i = 1, 2, . . . , n.
Exercise 1.2.2 Show Theorem 1.2.1.
3
are linearly independent) if the equation
n
i=1
a i v i = 0 implies that a i = 0 for all
i = 1, 2, . . . , n. If there exists at least one a i = 0 such that
n
i=1
a i v i = 0, then S (or
that the vectors v 1 , v 2 , . . . , v n ) is called linearly dependent (LD).
Definition 1.1.6 Let V be a vector space. A basis for V is a linearly independent set
of vectors in V which spans V . We say that V is finite-dimensional if it has a finite
basis.
Theorem 1.1.2 (Invariance of dimension) Let V be a vector space with bases S =
{v 1 , v 2 , . . . , v n } and T = {w 1 , w 2 , . . . , w m }. Then it follows that m = n.
Because of the invariance of dimension (Theorem 1.1.2), one can define the dimension of a vector space.
Definition 1.1.7 The dimension dim K (V ) of a finite-dimensional vector space V
(over K ) is the number of elements in a basis of V .
1.2 Linear Transformations
Here, we recall the concepts of linear transformation and linear operator.
Definition 1.2.1 Let V and W be two vectors spaces over the same field K . A
function T : V −→ W is a linear transformation from V into W if the following
hold:
(T1) ∀ u, v ∈ V , T (u + v) = T (u) + T (v);
(T2) ∀ v ∈ V , and ∀ a ∈ K , T (av) = aT (v).
In Linear Algebra, isomorphic vector spaces are essentially the same.
Definition 1.2.2 Let V and W be two vectors spaces. We say that V and W
are isomorphic, written V ∼ = W , if there exists a bijective linear transformation
T : V −→ W .
Exercise 1.2.1 Show that if V and W are finite-dimensional vector spaces then
V ∼ = W if and only if dim K (V ) = dim K (W ).
Theorem 1.2.1 Let V and W be vector spaces and let v 1 , v 2 , . . . , v n be a basis
of V . Given any list of vectors w 1 , w 2 , . . . , w n in W , there exists a unique linear
transformation T : V −→ W such that T (v i ) = w i for all i = 1, 2, . . . , n.
Exercise 1.2.2 Show Theorem 1.2.1.