Ax = b must have a solution for eac In Exercises 33 and 34, T is a linear transformation from R2 into R2. Show that T is invertible and find a formula for T- 33. T(x1, x₂) = (-5x1 + 9x2,4x17x2) 34. T(x₁, x₂) = (6x₁ - 8x2, -5x₁ + 7x₂) 35. Let T: R" → R" be an invertible linear transformation. Ex- plain why T is both one-to-one and onto R". Use equations (1) and (2). Then give a second explanation using one or more theorems. 36. Let T be a linear transformation that maps R" onto R". Show that T- exists and maps R" onto R". Is T-¹ also one-to- one? 37. Suppose T and U are linear transformations from R" to R" such that T (UX) = x for all x in R". Is it true that U(TX) = x for all x in R"? Why or why not? 38. Suppose a linear transformation T: R" → R" has the prop- erty that T(u) = T(v) for some pair of distinct vectors u and v in R". Can T map R" onto R"? Why or why not? FOTO 39. Let T: R" → R" be an invertible linear transformation, and let S and U be functions from R" into R" such that S (T(x)) = = x and U (T(x)) = x for all x in R". Show that U(v) = S(v) for all v in R". This will show that I has a unique inverse, as asserted in Theorem 9. [Hint: Given any v in R", we can write v = T(x) for some x. Why? Compute S(v) and U(v).] TUSHO lepps sdr 40. Suppose T and S satisfy the invertibility equations (1) and (2), where T is a linear transformation. Show directly that S is a linear transformation. [Hint: Given u, v in R", let x = S(u), y = S(v). Then 7(x) = u, T(y) = v. Why? Apply S to both sides of the equation T(x) + T(y) = T(x+y). Also, consider T(cx) = cT (x).] Hous 8:21,02 siding| houlda 21 nousups her word. sood idio c. Use ber of the coeffici Exercises 42-44 show how trix A to estimate the accur If the entries of A and ba and if the condition numb positive integer), then the usually be accurate to at l 42. [M] Find the conditie Construct a random Then use your matri of Ax= b. To how m the number of digits and report how man used in place of the e 43. [M] Repeat Exercise 44. [M] Solve an equatic column of the invers A = 1 1/2 1/3 1/4 1/5 SG 1/2 1/3 1/4 1/5 1/6 How many digits in correct? Explain. [N 56700,-88200, 441 45. [M] Some matrix pr mand to create Hilbe use an inverse comm order or larger Hill what you find. Mastering: Review SOLUTIONS TO PRACTICE PROBLEMS 1. The columns of A are obviously linearly depende tiples of column 1. Hence A cannot want Theorem.