a basis for r2 Consider an arbitrary set of vectors in R, a, b, and c. If a and b are a basis, we can find numbers a and a% such that c=ara+a2b. Let b b2 C2 Then C1=a1a1+a2b1, C2=a1a2+ The solutions to this pair of equations are a1 C2-a2C1 b2- b1a2 This gives a unique solution unless(a1b2-b1a2)=0. If(a1b2-b1a2)=0, then a1/a2=61/b2, which means that b is just a multiple of a. This returns us to our original condition, that a and b point in different direction. The implication is that if a and b are any pair of vectors for which the denominator in o is not zero, then any other vector c can be formed as a unique linear combination of and b. The basis of a vector space is not unique, since any set of vectors that satisfy the definition will do. But for any particular basis, there is only one linear combination of them that will produce another particular vector in the vector space. 3.3 Linear Dependence As the preceding should suggest, k vectors are required to form a basis for R However it is not every set of k vectors will suffices. As we see, to form a basis we require that this k vectors to be linearly independent Definition: A sets of vectors is linearly dependent if any one of the vectors in the set can bea basis for R 2 . Proof: Consider an arbitrary set of vectors in R 2 , a, b, and c. If a and b are a basis, we can find numbers α1 and α2 such that c = α1a + α2b. Let a =  a1 a2  , b =  b1 b2  , and c =  c1 c2  . Then c1 = α1a1 + α2b1, c2 = α1a2 + α2b2. The solutions to this pair of equations are α1 = b2c1 − b1c2 a1b2 − b1a2 , (1) α2 = a1c2 − a2c1 a1b2 − b1a2 . (2) This gives a unique solution unless (a1b2 − b1a2) = 0. If (a1b2 − b1a2) = 0, then a1/a2 = b1/b2, which means that b is just a multiple of a. This returns us to our original condition , that a and b point in different direction. The implication is that if a and b are any pair of vectors for which the denominator in () is not zero, then any other vector c can be formed as a unique linear combination of a and b. The basis of a vector space is not unique, since any set of vectors that satisfy the definition will do. But for any particular basis, there is only one linear combination of them that will produce another particular vector in the vector space. 3.3 Linear Dependence As the preceding should suggest, k vectors are required to form a basis for R k . However it is not every set of k vectors will suffices. As we see, to form a basis we require that this k vectors to be linearly independent. Definition: A sets of vectors is linearly dependent if any one of the vectors in the set can be 9