Math 300
Notes for Section 4.5
1. (Definition of a Basis). Let Vbe a vector space. A set SDfEv1;Ev2;:::;Evngof vectors in Vis
called a basis for Vif the following conditions hold:
(a) Sspans V.
(b) Sis linearly independent.
2. Theorem (Spanning and Linear Dependence). If a set SDfEv1;Ev2;:::;Evngspans Vthen
every set Tcontaining more than nvectors in Vis linearly dependent.
3. (All Bases for VHave the Same Size). If Vhas a basis with nvectors then every basis for V
has nvectors.
4. (Definition of Dimension). Let Vbe a vector space with a basisconsisting of nvectors. We call
the number nthe dimension of V, and write dim.V / Dn.IfVDf
E
0g, the dimension of Vis
defined to be zero.
5. (Finite Dimensional and Infinite Dimensional Vector Spaces). Not every vector space Vhas
a (finite) basis. If Vhas a (finite) basis or VDf
E
0g, then Vis called finite dimensional. Otherwise,
Vis called infinite dimensional.
6. Theorem (Basis Test When We Know That dim.V / Dn). Suppose Vis a vector space with
dim.V / Dn, and let SDfEv1;Ev2;:::;Evkgbe a set of kvectors in V. Then:
(a) If k<n, then Sdoes not span V.
(b) If k>n, then Sis linearly dependent.
(c) If Sis linearly independent and kDn, then Sspans V.
(d) If Sspans Vand kDn, then Sis linearly independent.
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