Chapter 3 Linear transformations

ker(T) ={p(x): p’(x)=0}={ all constant polynomials}. So nullity(T)=1.Also rang(T)= P n −1, so rank(T)=n.Note that rank(T)+nullity(T)=n+1=dim(P n).3.1.7 Theorem(Dimension Theorem for linear transformations})Let T:V → W be a linear transformation from an n-dimensional space V to an m-dimensional space W, thenrank(T)+nullity(T)=n.3.1.8 Theorem Let T: V→ W be a linear transformation. ThenT is injective if and only if ker(T)={0}.Exercise 3.11. Determine which of the following mappings are linear transformations.(1) T: P n→ R n , where T sends a polynomial p= a nx n +a −1to the vector T(p)=( a 0, a 1+ a 0,. . . , a n+a n −1+… +a 1+ a 0).nx −1n +… +a 1x+ a 0(2) T: C[0,1] → R, where for each f(x) in C[0,1],T(f)= ∫ 1 0f ( x)h(x)dx ,where h(x) is a fixed continuous function.