Chapter 3 Linear transformations

More documents

Recommendations

Info

Ker(T)0VWR(T)VW3.1.5 Theorem Let T:V → W be a linear transformation. Then R(T) is a subspace of W,and ker(T) is a subspace of V.3.1.6 Example(1) Let T A: R n → R m be the linear transformation defined by T A(u)=Au, whereA is an m× n matrix. Then ker(T A) ={ u: T A(u)=0} ={u: Au=0}, which is actually thenullspace of A . Also R(T A)={ Au: u is an arbitrary vector in R n }, so it is the columnspace of A.(2 ) Let V be a vector space. Define F: V → V by F(u)=3u, for all u in V. ThenF is a linear transformation. What is ker(F) and R(F) ?(3) Let T: P n→ P n −1be the derivative function. Then
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 <strong>transformations</strong>})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 <strong>transformations</strong>.(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.
Page 1 and 2: The Landmark Tshipi Acquisition & C
Page 3: Thus P 2 is isomorphic to R 3 , and
Page 7 and 8: Also (1,0 )=T(1,0,0)=T(e 1 ), (1,1)
Page 9 and 10: Thus [T]S, B=A=[ F(e 1) F(e 2) …
Page 11 and 12: Solution: By the above resultP=[[1]
Page 13 and 14: is invertible and A −1is similar

Chapter 3 Linear transformations

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?