Linear Algebra

112 Chapter Two. Vector SpacesIIIBasis and DimensionThe prior section ends with the statement that a spanning set is minimal when itis linearly independent and a linearly independent set is maximal when it spansthe space. So the notions of minimal spanning set and maximal independentset coincide. In this section we will name this idea and study its properties.III.1 Basis1.1 Definition A basis for a vector space is a sequence of vectors that forma set that is linearly independent and that spans the space.We denote a basis with angle brackets 〈 ⃗ β 1 , ⃗ β 2 , . . .〉 to signify that this collectionis a sequence ∗ — the order of the elements is significant. (The requirementthat a basis be ordered will be needed, for instance, in Definition 1.13.)1.2 Example This is a basis for R 2 .( (2 1〈 , 〉4)1)It is linearly independent( ) ( ) (2 1 0c 1 + c4 2 =1 0)and it spans R 2 .=⇒ 2c 1 + 1c 2 = 04c 1 + 1c 2 = 0=⇒ c 1 = c 2 = 02c 1 + 1c 2 = x4c 1 + 1c 2 = y=⇒ c 2 = 2x − y and c 1 = (y − x)/21.3 Example This basis for R 2 ( (1 2〈 , 〉1)4)differs from the prior one because the vectors are in a different order.verification that it is a basis is just as in the prior example.The1.4 Example The space R 2 has many bases. Another one is this.( (1 0〈 , 〉0)1)The verification is easy.∗ More information on sequences is in the appendix.