Problem 1E: Show that each of the following are linear operators on 2 . Describe geometrically what each linear... Problem 2E: Let L be the linear operator on 2 defined by L(x)=(x1cosx2sin+x2cos)T Express x1 , x2 , and L(x) in... Problem 3E: Let a be a fixed nonzero vector in 2 . A mapping of the form L(x)=x+a is called a translation. Show... Problem 4E: Let L: 22 be a linear operator. If L((1,2)T)=(2,3)T and L((1,1)T)=(5,2)T find the value of L((7,5)T)... Problem 5E: Determine whether the following are linear transformations from 3 into 2 . (a) L(x)=(x2+x3)T (b)... Problem 6E: Determine whether the following are linear transformations from 2 into 3 . (a) L(x)=(x1,x2,1)T (b)... Problem 7E: Determine whether the following are linear operators on nn . (a) L(A)=2A (b) L(A)=AT (c) L(A)=A+1... Problem 8E: Let C be a fixed nn matrix. Determine whether the following are linear operators on nn . (a)... Problem 9E: Determine whether the following are linear transformations from P2 to P3 . (a) L(p(x))=xp(x) (b)... Problem 10E: For each fC[0,1] , define L(f)=F , where F(x)= 0 x f(t)dt0x1 Show that L is a linear operator on... Problem 11E: Determine whether the following are linear transformations from C[0,1] into 1 : (a) L(f)=f(0) (b)... Problem 12E: Use mathematical induction to prove that if L is a linear transformation from V to W, then... Problem 13E: Let {v1,...,vn} be a basis for a vector space V, and let L1 and L2 be two linear transformations... Problem 14E: Let L be a linear operator on 1 and let a=L(1) . Show that L(x)=ax for all x1 . Problem 15E: Let L be a linear operator on a vector space V. Define Ln,n1 , recursively by L1=L Lk+1(v)=L(Lk(v))... Problem 16E: Let L1:UV and L2:VW be a linear transformations, and let L=L2L1 be the mapping defined by... Problem 17E: Determine the kernel and range of each of the following linear operators on 3 : (a) L(x)=(x3,x2,x1)T... Problem 18E: Let S be the subspace of 3 spanned by e1 and e2 . For each linear operator L in Exercise 17, find... Problem 19E: Find the kernel and range of each of the following linear operators on P3 : (a) L(p(x))=xp(x) (b)... Problem 20E: Let L:VW be a linear transformation, and let T be a subspace of W. The inverse image of T, denoted... Problem 21E: A linear transformation L:VW is said to be onetoone if L(v1)=L(v2) implies that v1=v2 (i.e., no two... Problem 22E: A linear transformation L:VW is said to be map V onto W if L(V)=W . Show that the linear... Problem 23E: Which of the operators defined in Exercise 17 are onetoone? Which maps 3 onto 3 ? Problem 24E: Let A be a 22 matrix, and let LA be the linear operator defined by LA(x)=Ax Show that (a) LA maps 2... Problem 25E: Let D be the differentiation operator on P3 , and let S={pP3|p(0)=0} Show that (a) D maps P3 onto... format_list_bulleted