Homework Sourse2 Show that the set of vectors XA2 XAXB XB2 is a bassis
2. Show that the set of vectors { (X-A)2 , (X-A)(X-B) , (X-B)2 } is a bassis of P2
Solution
First, we prove that Span{(X-A)2, (X-A)(X-B), (X-B)2} = P2. Indeed, every vector in Span{(X-A)2, (X-A)(X-B), (X-B)2 } has the form a + bt + ct2 for some a, b, c R; this is exactly the form of a general element of P2. Next, we prove that {(X-A)2, (X-A)(X-B), (X-B)2} is linearly independent. Let us assume that a(X-A)2 + b(X-A)(X-B) + c(X-B)2= 0. Then the coefficients of this polynomial are zero, which implies a = b = c = 0.
Thus {(X-A)2, (X-A)(X-B), (X-B)2 are linearly independent. Therefore, the set of vectors { (X-A)2 , (X-A)(X-B) , (X-B)2 } is a basis of P2
