2270-2, LINEAR ALGEBRA, DEFINITIONS IN CHAPTER 4, 02/15/2005
Definition 1. A linear space V is a set endowed with a rule for addition and scalar multiplication, (if
f, g ∈ V , then f + g, kf ∈ V for all k ∈ R) which satisfy the following 8 properties: for all f, g, h ∈ V and
all scalars c, k ∈ R,
(1) Associativity under +: (f + g) + h = f + (g + h)
(2) Commutativity under +: f + g = g + f
(3) Neutral element under +: there exists a neutral element n ∈ V such that f + n = f for all f ∈ V .
This n is unique and denoted by 0.
(4) Inverse element under +: for each f ∈ V , there exists g ∈ V such that f + g = 0. This g is unique
and denoted by (−f ).
(5) Scalar distributive law I: k(f + g) = kf + kg
(6) Scalar distributive law II: (c + k)f = cf + kf
(7) Associativity by scalars: c(kf ) = (ck)f
(8) 1f = f
Definition 2. A subset W of a linear space V is called a subspace of V if
(1) 0 ∈ W (The neutral element of V is in W )
(2) W is closed under addition
(3) W is closed under scalar multiplication.
((2) and (3) can be summarized by saying that W is closed under linear combinations.)
Definition 3. h Span, linear dependence, basis, coordinates i
Consider the elements f1 , . . . , fn in a linear space V .
(1) We say that f1 , . . . fn span V , if every f ∈ V can be expressed as a linear combination of f1 , . . . fn .
(2) We say that f1 , . . . fn are linearly independent, if the equation c1 f1 + · · · + cn fn = 0 has only
the trivial relation (solution) c1 = · · · = cn = 0.
(3) We say that elements f1 , . . . fn are a basis of V if they span V and are linearly independent. This
means that every f ∈ V can be written as a linear combination f = c1 f1 + · · · + cn fn and the coefficients
c1 , . . . , cn are called the coordinates of f 
w.r.t the basis B = (f1 , . . . , fn ).
c1
.

The B-coordinate vector of f is  .. 
 and is denoted by [f ]B .
cn
 
c1
.
n

The B-coordinate transformation of T : V → R is T (f ) = [f ]B =  .. 
.
cn
Definition 4. If a linear space V has a basis with n elements, then define its dimension dim(V ))=n.
If n is finite, (i.e. V has finitely many elements in a basis,) then we say V is finite dimensional.
Otherwise, we say V is infinite dimensional.
We discuss lots of important examples and properties of linear spaces and subspaces in class.
Date: February 15, 2005.
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