Exercise 9.2. Let V = Π., R = 1 {(a1, · · ·‚ an, .) | a1, ..., An, ... Є R} and R {linear transformations T : V → V}. Define the linear transformation To : V → V by To(a1, a2,, an,) = (a2, a3, , an-1, ). · Show that To is a left zero divisor of R but not a right zero divisor of R. = L(V) =

Let T be a linear transformation from R2 into R2 such that T(2,0)=(1,1) and T(0,3)=(3,3). Find T(1,1) and T(0,1)

Let T be a linear transformation from R3 into R such that T(1,1,1)=1, T(1,1,0)=2 and T(1,0,0)=3. Find T(0,1,1)

Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).

Let T:RnRm be the linear transformation defined by T(v)=Av, where A=[30100302]. Find the dimensions of Rn and Rm.

Find a basis for R2 that includes the vector (2,2).

Let T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.

For the linear transformation from Exercise 45, let =45 and find the preimage of v=(1,1). 45. Let T be a linear transformation from R2 into R2 such that T(x,y)=(xcosysin,xsin+ycos). Find a T(4,4) for =45, b T(4,4) for =30, and c T(5,0) for =120.

Let T be a linear transformation from R2 into R2 such that T(x,y)=(xcosysin,xsin+ycos). Find a T(4,4) for =45, b T(4,4) for =30, and c T(5,0) for =120.

A translation in R2 is a function of the form T(x,y)=(xh,yk), where at least one of the constants h and k is nonzero. (a) Show that a translation in R2 is not a linear transformation. (b) For the translation T(x,y)=(x2,y+1), determine the images of (0,0,),(2,1), and (5,4). (c) Show that a translation in R2 has no fixed points.

For the linear transformation from Exercise 38, find a T(0,1,0,1,0), and b the preimage of (0,0,0), c the preimage of (1,1,2). Linear Transformation Given by a Matrix In Exercises 33-38, define the linear transformation T:RnRmby T(v)=Av. Find the dimensions of Rnand Rm. A=[020201010112221]

For the linear transformation from Exercise 33, find a T(1,1), b the preimage of (1,1), and c the preimage of (0,0). Linear Transformation Given by a Matrix In Exercises 33-38, define the linear transformations T:RnRm by T(v)=Av. Find the dimensions of Rn andRm. A=[0110]

Exercise 273. Show that projw is a linear transformation and that (projw)2 = projw.