80 CHAPTER 1 Linear Equations in Linear Algebra c. The standard matrix of a linear transformation from R² to R2 that reflects points through the horizontal axis, the vertical axis, or the origin has the form [ 2] where a and d are ±1. d. A mapping T:R". →R" is one-to-one if each vector in R" maps onto a unique vector in R". e. If A is a 3 x 2 matrix, then the transformation x Ax cannot map R2 onto R³. In Exercises 25-28, determine if the specified linear transforma- tion is (a) one-to-one and (b) onto. Justify each answer. 25. The transformation in Exercise 17 26 The transformation in Exercise 2 27. The transformation in Exercise 19 28. The transformation in Exercise 14 In Exercises 29 and 30, describe the possible echelon forms of the standard matrix for a linear transformation T. Use the notation of Example 1 in Section 1.2. 29. T: R³ R4 is one-to-one. 30. T: RR³ is onto. 31. Let T: R" → R" be a linear transformation, with A its standard matrix. Complete the following statement to make it true: "T is one-to-one if and only if A has - pivot columns." Explain why the statement is true. [Hint: Look in the exercises for Section 1.7.] - → 32. Let T: R" Rm be a linear transformation, with A its standard matrix. Complete the following statement to make it true: "T maps R" onto R" if and only if A has pivot columns." Find some theorems that explain why the statement is true. 33. Verify the uniqueness of A in Theorem 10. Let T:R" → R" be a linear transformation such that T(x)=Bx for some WEB 34. Why is the question "Is the linear transformation T onto?" an existence question? 35. If a linear transformation T: R"R" maps R" onto R™, can you give a relation between m and n? If T is one-to-one, what can you say about m and n? mxn matrix B. Show that if A is the standard matrix for T, then A= B. [Hint: Show that A and B have the same columns.] 36. Let S : RP→ R" and T: R" → R" be linear transforma- tions. Show that the mapping x→ T(S(x)) is a linear trans- formation (from RP to R"). [Hint: Compute T(S(cu + dv)) for u, v in RP and scalars c and d. Justify each step of the computation, and explain why this computation gives the desired conclusion.] [M] In Exercises 37-40, let 7 be the linear transformation whose standard matrix is given. In Exercises 37 and 38, decide if T is a one-to-one mapping. In Exercises 39 and 40, decide if T maps R5 onto R5. Justify your answers. 37. 39. 40. -5 8 4-9 -3-2 ? тотти афро 4 -7 -7 3 6-8 5 12 -8 10 -5 4 7 3-4 5 -3 -5 3-5 9 13 13 5 6 -1 -7 -6 14 15 -8-9 12-5-9 -5-6-8 9 8 14 15 2 11 55 10 -8 -9 14 4 2-6 6 -6 -7 SOLUTION TO PRACTICE PROBLEMS X₂ 38. 75 4 -9] 6 16-4 10 12 8 12 7 -8-6-2 5 1. Follow what happens to e, and e₂. See Figure 5. First, e, is unaffected by the shear and then is reflected into -e₁. So T (e) = -e₁. Second, e2 goes to e2-.5e by the shear transformation. Since reflection through the x2-axis changes e, into -e, and