Answer: In Exercises 25–28, determine if the specified

Problem 1E In Exercises 1–10, assume that T is a linear transformation. Find the standard matrix of T.

Problem 3E In Exercises 1–10, assume that T is a linear transformation. Find the standard matrix of T. T : ?2 ? ?2 is a vertical shear transformation that maps e1 into e1 – 3e2, but leaves e2 unchanged.

Problem 40E [M] In Exercises 37–40, let T 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 ?5 onto ?5. Justify your answers.

Problem 2E In Exercises 1–10, assume that T is a linear transformation. Find the standard matrix of T. where e1, e2, and e3 are the columns of the 3 × 3 identity matrix.

Problem 7E In Exercises 1–10, assume that T is a linear transformation. Find the standard matrix of T. T : ?2 ? ?2 first rotates points through –3?/4 radians (clockwise) and then reflects points through the horizontal x1-axis.

Problem 6E In Exercises 1–10, assume that T is a linear transformation. Find the standard matrix of T. T : ?2 ? ?2 rotates points (about the origin) through –3?/2 radians (clockwise).

Problem 4E In Exercises 1–10, assume that T is a linear transformation. Find the standard matrix of T. T : ?2 ? ?2 is a horizontal shear transformation that leaves e1 unchanged and maps e2 into e2 – 2e1.

Problem 9E In Exercises 1–10, assume that T is a linear transformation. Find the standard matrix of T. T : ?2 ? ?2 first reflects points through the horizontal x1-axis and then rotates points –?/2 radians

Problem 5E In Exercises 1–10, assume that T is a linear transformation. Find the standard matrix of T. T : ?2 ? ?2 rotates points (about the origin) through ?/2 radians (counterclockwise).

Problem 10E In Exercises 1–10, assume that T is a linear transformation. Find the standard matrix of T. T : ?2 ? ?2 first reflects points through the horizontal x1-axis and then reflects points through the line x2 = x1

Problem 12E Show that the transformation in Exercise 10 is merely a rotation about the origin. What is the angle of the rotation? Exercise 10: Assume that T is a linear transformation. Find the standard matrix of T. T : ?2 ? ?2 first reflects points through the horizontal x1-axis and then reflects points through the line x2 = x1

Problem 13E Let T : ?2 ? ?2 be the linear transformation such that T (e1) and T (e2) are the vectors shown in the figure. Using the figure, sketch the vector T (2, 1)

Problem 15E In Exercises 15 and 16, fill in the missing entries of the matrix, assuming that the equation holds for all values of the variables.

Problem 16E In Exercises 15 and 16, fill in the missing entries of the matrix, assuming that the equation holds for all values of the variables.

Problem 17E In Exercises 17–20, show that T is a linear transformation by finding a matrix that implements the mapping. Note that x1, x2,… are not vectors but are entries in vectors.

Problem 11E A linear transformation T : ?2 ? ?2 first reflects points through the x1-axis and then reflects points through the x2- axis. Show that T can also be described as a linear transformation that rotates points about the origin. What is the angle of that rotation?

Problem 14E Let T : ?2 ? ?2 be a linear transformation with standard matrix A = [a1 a2] where a1 and a2 are shown in the figure at the top of column 2. Using the figure, draw the image of under the transformation T.

Problem 18E In Exercises 17–20, show that T is a linear transformation by finding a matrix that implements the mapping. Note that x1, x2,… are not vectors but are entries in vectors..

Problem 21E Let T : ?2 ? ?2 be a linear transformation such that . Find x such that T (x) = (3, 8).

Problem 19E In Exercises 17–20, show that T is a linear transformation by finding a matrix that implements the mapping. Note that x1, x2,… are not vectors but are entries in vectors.

Problem 20E In Exercises 17–20, show that T is a linear transformation by finding a matrix that implements the mapping. Note that x1, x2,… are not vectors but are entries in vectors.

Problem 23E In Exercises 23 and 24, mark each statement True or False. Justify each answer. a. A linear transformation T : ?n ? ?m is completely determined by its effect on the columns of the n × n identity matrix. b. If T : ?2 ? ?2 rotates vectors about the origin through an angle ?, then T is a linear transformation. c. When two linear transformations are performed one after another, the combined effect may not always be a linear transformation. d. A mapping T : ?n ? ?m is onto ?m if every vector x in ?n maps onto some vector in ?m. e. If A is a 3 × 2 matrix, then the transformation x ? Ax cannot be one-to-one.

Problem 25E In Exercises 25–28, determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer. The transformation in Exercise 17 Exercises 17: Show that T is a linear transformation by finding a matrix that implements the mapping. Note that x1, x2,… are not vectors but are entries in vectors.

Problem 22E Let T : ?2 ? ?3 be a linear transformation with Find x such that

Problem 24E In Exercises 23 and 24, mark each statement True or False. Justify each answer. a. If A is a 4 × 3 matrix, then the transformation x ? Ax maps ?3 on to ?4. b. Every linear transformation from ?n to ?m is a matrix transformation. c. The columns of the standard matrix for a linear transformation from ?n to ?m are the images of the columns of the n × n identity matrix under T. d. A mapping ?n ? ?m is one-to-one if each vector in ?n maps onto a unique vector in ?m. e. The standard matrix of a horizontal shear transformation from ?2 to ?2 has the form , where a and d are ±1.

Problem 26E In Exercises 25–28, determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer. The transformation in Exercise 2 Exercises 2: Assume that T is a linear transformation. Find the standard matrix of T. where e1, e2, and e3 are the columns of the 3 × 3 identity matrix.

Problem 27E In Exercises 25–28, determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer. The transformation in Exercise 19 Exercises 19: Show that T is a linear transformation by finding a matrix that implements the mapping. Note that x1, x2,… are not vectors but are entries in vectors.

Problem 30E 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 T : ?4 ? ?3 is onto. Example 1: The following matrices are in echelon form. The leading entries ( ? ) may have any nonzero value; the starred entries (*) may have any value (including zero). The following matrices are in reduced echelon form because the leading entries are 1’s, and there are 0’s below and above each leading 1.

Problem 28E In Exercises 25–28, determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer. The transformation in Exercise 14 Exercise 14: Let T : ?2 ? ?2 be a linear transformation with standard matrix A = [a1 a2] where a1 and a2 are shown in the figure at the top of column 2. Using the figure, draw the image of under the transformation T.

Problem 31E Let T : ?n? ?m 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.]

Problem 34E be linear transformations. Show that the mapping is a linear transformation . [Hint: Compute and scalars c and d.Justify each step of the computation, and explain why this computation gives the desired conclusion.]

Problem 36E Why is the question “Is the linear transformation T onto?” an existence question?

Problem 29E 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 is one-to-one. Example 1: The following matrices are in echelon form. The leading entries ( ? ) may have any nonzero value; the starred entries (*) may have any value (including zero). The following matrices are in reduced echelon form because the leading entries are 1’s, and there are 0’s below and above each leading 1.

Problem 35E If a linear transformation can you give a relation between m and n? If T is one-to-one, what can you say about m and n?

Problem 32E Let T : ?n? ?m be a linear transformation, with A its standard matrix. Complete the following statement to make it true: “T maps ?n onto ?m if and only if A has __________ pivot columns.” Find some theorems that explain why the statement is true.

Problem 33E Verify the uniqueness of A in Theorem 10. Let T : ?n? ?m be a linear transformation such that T (x) = Bx for some m × n 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.] Theorem 10: Let T : ?n? ?m be a linear transformation. Then there exists a unique matrix A such that for all x in ?n In fact, A is the m × n matrix whose j th column is the vector where ej is the j th column of the identity matrix in ?n:

Problem 37E [M] In Exercises 37–40, let T 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 ?5 onto ?5. Justify your answers.

Problem 39E [M] In Exercises 37–40, let T 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 ?5 onto ?5. Justify your answers.

Problem 38E [M] In Exercises 37–40, let T 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 ?5 onto ?5. Justify your answers.