33 80 CHAPTER 1 Linear Equations in Linear Algebrac. The standard matrix of a linear transformation from R2to 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 inR" maps onto a unique vector in R".e. If A is a 3 x 2 matrix, then the transformation x + Axcannot 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 1726 The transformation in Exercise 227. The transformation in Exercise 1928. The transformation in Exercise 14In Exercises 29 and 30, describe the possible echelon forms of thestandard matrix for a linear transformation T. Use the notation ofExample 1 in Section 1.2.29. T: R³ R4 is one-to-one.30. T: RR³ is onto.31. Let T: R" → Rm be a linear transformation, with A itsstandard matrix. Complete the following statement to makeit true: "T is one-to-one if and only if A has pivotcolumns." Explain why the statement is true. [Hint: Look inthe exercises for Section 1.7.]32. Let T: R → R" be a linear transformation, with A itsstandard matrix. Complete the following statement to makeit true: "T maps R" onto R" if and only if A haspivot columns." Find some theorems that explain why thestatement is true.33. Verify the uniqueness of A in Theorem 10. Let T: R" → Rmbe a linear transformation such that T(x) = Bx for someWEBx2A·x₁34. Why is the question "Is the linear transformation T onto?"an existence question?35. If a linear transformation T: R"→ R" maps R" onto Rcan 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 forT, then A= B. [Hint: Show that A and B have the samecolumns.]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 thecomputation, and explain why this computation gives thedesired conclusion.][M] In Exercises 37-40, let T be the linear transformation whosestandard matrix is given. In Exercises 37 and 38, decide if T is aone-to-one mapping. In Exercisesand 40, decide if T maps R$onto R5. Justify your answers.37.39.40.-5 10 -58 3 -4474 -9 5 -3-3 -2 544 -7 3 76-8 5 12 -8-7 10 -8 -9 143-5 4 2-66 -6 -7 3-5913 514 15 -7-8 -9 12-5 -6-813 14 156 -1-6 4-5-99 811x₁Shear transformationFIGURE 5 The composition of two transformations.38.57pohoa lan5 4 -9106 16 -48 1212-8 -6 -2SOLUTION TO PRACTICE PROBLEMS1. Follow what happens to e, and e2. See Figure 5. First, e₁ is unaffected by the shearand then is reflected into -e₁. So T (e₁) = -e₁. Second, e2 goes to e2 - .5e, by theshear transformation. Since reflection through the x2-axis changes e, into -e₁ and[1]1'-0'7mning o+₂woda aus75Reflection through the x-axis

Theorem 10 states that if T: R^n -> R^m is a linear transformation, and A is the standard matrix…

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 [82] 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: R¹ R³ 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" → R" 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 34. Why is the question "Is the linear transformation T ontc an existence question? mx n matrix B. Show that if A is the standard matrix T, then A= B. [Hint: Show that A and B have the san columns.] 35. If a linear transformation T: R"R" maps R" onto can you give a relation between m and n? If T is one-to-o what can you say about m and n? 36. Let S : RPR" and T: R"→ R" be linear transform tions. Show that the mapping x→ T(S(x)) is a linear tram formation (from RP to R"). [Hint: Compute T(S(cu + d for u, v in RP and scalars c and d. Justify each step of t computation, and explain why this computation gives t desired conclusion.] [M] In Exercises 37-40, let T be the linear transformation who standard matrix is given. In Exercises 37 and 38, decide if T is one-to-one mapping. In Exercises 39 and 40, decide if 7 maps onto R5. Justify your answers. 37. 39. -5 10 -5 8 3-4 4-9 5 -3 -3 -2 5 4 734 57 4-7 37 6-8 5 12 -8 -7 10 -8 -9 14 3-5 4 2-6 nomi -5 6 -6 -7 3 9 13 5 6 -1] 14 15 -7 -6 4 40. -8-9 12 -5 -9 olg-5 -6 -8 9 8 13 14 15 2 11 38. 7 5 4 -9 10 6 16-4 12 8 12 7 -8-6-2 5