21 1.9 EXERCISESIn Exercises 1-10, assume that T is a linear transformation. Findthe standard matrix of T.1. T : R² → R4, 7(e₁) = (3, 1, 3, 1) and 7(e₂) = (-5,2,0,0),where e₁ = (1, 0) and e₂ = (0, 1).:-2.7: R³ R², T(e₁) = (1,3), 7(e₂) =(4,-7), andT(e3)= (-5,4), where e₁, e2, e3 are the columns of the3 x 3 identity matrix.3. T: R2 R2 rotates points (about the origin) through 37/2uradians (counterclockwise). de m→4. T R2 R2 rotates points (about the origin) through -л/4radians (clockwise). [Hint: T(e₁) = (1/√2, -1/√2).]www.NRICS5. T: R² R2 is a vertical shear transformation that maps e₁into e₁ - 2e₂ but leaves the vector e₂ unchanged.ody no render wall on6. T: R²R2 is a horizontal shear transformation that leavese₁ unchanged and maps e₂ into e₂ + 3e₁.→7. T: R² R² first rotates points through -3/4 radian(clockwise) and then reflects points through the horizontalx₁-axis. [Hint: T(e₁) = (-1/√2, 1/√2).]8. T: R² → R² first reflects points through the horizontal x₁-axis and then reflects points through the line x2 = X1.9. T: R² R² first performs a horizontal shear that trans-forms e2 into e2 - 2e₁ (leaving e, unchanged) and then re-flects points through the line x₂ = -X₁.0. T: R² R2 first reflects points through the vertical x2-axisand then rotates points л/2 radians.1. A linear transformation T: R2 R2 first reflects pointsthrough the x₁-axis and then reflects points through the x2-axis. Show that T can also be described as a linear transfor-mation that rotates points about the origin. What is the angleof that rotation?2. Show that the transformation in Exercise 8 is merely a rota-tion about the origin. What is the angle of the rotation?3. Let T: R2 R2 be the linear transformation such that T(e₁)and T(₂) are the vectors shown in the figure. Using thefigure, sketch the vector T(2, 1).T(e,)x2T(e₂)X14. Let T: R² → R² be a linear transformation with standardmatrix A = [a₁ a2], where a, and a2 are shown in theunder thefigure. Using the figure, draw the image of [3]transttransformation T.anol d16.? ???15. ??TIn Exercises 15 and 16, fill in the missing entries of the matrix,assuming that the equation holds for all values of the variables.3x₁ - 2x34x1x1 - x₂ + x3th?X1X2I-?x2?? ??ift?X113]-[=?x2.X3a2x1 - x2-2x1 + x₂X1X1a₁In Exercises 17-20, show that T is a linear transformation byfinding a matrix that implements the mapping. Note that x₁, x2,...are not vectors but are entries in vectors.17. T(X1, X2, X3, X4) = (0, x₁ + x2, x2 + x3, x3 + x4)18./T(x₁, x2) = (2x2 - 3x1, x₁ - 4x2, 0, x₂)19. T(X₁, X2, X3) = (x1 - 5x2 + 4x3, x2 - 6x3)20. T(X1, X2, X3, X4) = 2x1 + 3x3 - 4x4 (T: R4 → R)21. Let T: R2 R2 be a linear transformation such thatT(x₁, x₂) = (x₁ + x2, 4x₁ + 5x2). Find x such that T(x) =(3,8).22. Let T R² R³ be a linear transformation such thatT(x₁, x2) = (x₁ - 2x2, -x1 + 3x2, 3x1 - 2x₂). Find x suchthat T(x) = (-1,4,9).0030In Exercises 23 and 24, mark each statement True or False. Justifyeach answer.m237.23. a. A linear transformation T: R" → R" is completely de-termined by its effect on the columns of the n x n identitymatrix.b. If T: R² R2 rotates vectors about the origin throughan angle , then T is a linear transformation.c. When two linear transformations are performed one afteranother, the combined effect may not always be a lineartransformation.d. A mapping T: R" → R" is onto Rm if every vector x inR"maps onto some vector in Rm.e. If A is a 3 x 2 matrix, then the transformation X→→ Axcannot be one-to-one.24. a. Not every linear transformation from R" to R" is a matrixtransformation.mb. The columns of the standard matrix for a linear transfor-mation from R" to R" are the images of the columns ofthe nxn identity matrix.

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21 1.9 EXERCISESIn Exercises 1-10, assume that T is a linear transformation. Findthe standard matrix of T.1. T : R² → R4, 7(e₁) = (3, 1, 3, 1) and 7(e₂) = (-5,2,0,0),where e₁ = (1, 0) and e₂ = (0, 1).:->2.7: R³ R², T(e₁) = (1,3), 7(e₂) =(4,-7), andT(e3)= (-5,4), where e₁, e2, e3 are the columns of the3 x 3 identity matrix.3. T: R2 R2 rotates points (about the origin) through 37/2uradians (counterclockwise). de m→>>4. T R2 R2 rotates points (about the origin) through -л/4radians (clockwise). [Hint: T(e₁) = (1/√2, -1/√2).]www.NRICS5. T: R² R2 is a vertical shear transformation that maps e₁into e₁ - 2e₂ but leaves the vector e₂ unchanged.ody no render wall on6. T: R²R2 is a horizontal shear transformation that leavese₁ unchanged and maps e₂ into e₂ + 3e₁.→>>7. T: R² R² first rotates points through -3/4 radian(clockwise) and then reflects points through the horizontalx₁-axis. [Hint: T(e₁) = (-1/√2, 1/√2).]8. T: R² → R² first reflects points through the horizontal x₁-axis and then reflects points through the line x2 = X1.9. T: R² R² first performs a horizontal shear that trans-forms e2 into e2 - 2e₁ (leaving e, unchanged) and then re-flects points through the line x₂ = -X₁.0. T: R² R2 first reflects points through the vertical x2-axisand then rotates points л/2 radians.1. A linear transformation T: R2 R2 first reflects pointsthrough the x₁-axis and then reflects points through the x2-axis. Show that T can also be described as a linear transfor-mation that rotates points about the origin. What is the angleof that rotation?2. Show that the transformation in Exercise 8 is merely a rota-tion about the origin. What is the angle of the rotation?3. Let T: R2 R2 be the linear transformation such that T(e₁)and T(₂) are the vectors shown in the figure. Using thefigure, sketch the vector T(2, 1).T(e,)x2T(e₂)X14. Let T: R² → R² be a linear transformation with standardmatrix A = [a₁ a2], where a, and a2 are shown in theunder thefigure. Using the figure, draw the image of [3]transttransformation T.anol d16.? ???15. ??TIn Exercises 15 and 16, fill in the missing entries of the matrix,assuming that the equation holds for all values of the variables.3x₁ - 2x34x1x1 - x₂ + x3th?X1X2I-?x2?? ??ift?X113]-[=?x2.X3a2x1 - x2-2x1 + x₂X1X1a₁In Exercises 17-20, show that T is a linear transformation byfinding a matrix that implements the mapping. Note that x₁, x2,...are not vectors but are entries in vectors.17. T(X1, X2, X3, X4) = (0, x₁ + x2, x2 + x3, x3 + x4)18./T(x₁, x2) = (2x2 - 3x1, x₁ - 4x2, 0, x₂)19. T(X₁, X2, X3) = (x1 - 5x2 + 4x3, x2 - 6x3)20. T(X1, X2, X3, X4) = 2x1 + 3x3 - 4x4 (T: R4 → R)21. Let T: R2 R2 be a linear transformation such thatT(x₁, x₂) = (x₁ + x2, 4x₁ + 5x2). Find x such that T(x) =(3,8).22. Let T R² R³ be a linear transformation such thatT(x₁, x2) = (x₁ - 2x2, -x1 + 3x2, 3x1 - 2x₂). Find x suchthat T(x) = (-1,4,9).0030In Exercises 23 and 24, mark each statement True or False. Justifyeach answer.m237.23. a. A linear transformation T: R&#34; → R&#34; is completely de-termined by its effect on the columns of the n x n identitymatrix.b. If T: R² R2 rotates vectors about the origin throughan angle , then T is a linear transformation.c. When two linear transformations are performed one afteranother, the combined effect may not always be a lineartransformation.d. A mapping T: R&#34; → R&#34; is onto Rm if every vector x inR&#34;maps onto some vector in Rm.e. If A is a 3 x 2 matrix, then the transformation X→→ Axcannot be one-to-one.24. a. Not every linear transformation from R&#34; to R&#34; is a matrixtransformation.mb. The columns of the standard matrix for a linear transfor-mation from R&#34; to R&#34; are the images of the columns ofthe nxn identity matrix.

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