Linear Algebra, 2020a

Section III. Basis and Dimension 121
III
Basis and Dimension
The prior section ends with the observation that a spanning set is minimal when
it is linearly independent and a linearly independent set is maximal when it spans
the space. So the notions of minimal spanning set and maximal independent set
coincide. In this section we will name this idea and study its properties.
III.1
Basis
1.1 Definition A basis for a vector space is a sequence of vectors that is linearly
independent and that spans the space.
Because a basis is a sequence, meaning that bases are different if they contain
the same elements but in different orders, we denote it with angle brackets
〈⃗β 1 , ⃗β 2 ,...〉. ∗ (A sequence is linearly independent if the multiset consisting of
the elements of the sequence is independent. Similarly, a sequence spans the
space if the set of elements of the sequence spans the space.)
1.2 Example This is a basis for R 2 .
( ) ( )
2 1
〈 , 〉
4 1
It is linearly independent
( ) ( ) ( )
2 1 0
c 1 + c 2 =
4 1 0
and it spans R 2 .
=⇒ 2c 1 + 1c 2 = 0
4c 1 + 1c 2 = 0
=⇒ c 1 = c 2 = 0
2c 1 + 1c 2 = x
4c 1 + 1c 2 = y
=⇒ c 2 = 2x − y and c 1 =(y − x)/2
1.3 Example This basis for R 2 differs from the prior one
( ) ( )
1 2
〈 , 〉
1 4
because it is in a different order. The verification that it is a basis is just as in
the prior example.
∗ More information on sequences is in the appendix.