Solution: Calculus In Exercises 6164, for the linear

Finding an Image and a Preimage In Exercises 18, use the function to find (a) the image of v and (b) the preimage of w. T(v1, v2) = (v1 + v2, v1 v2), v = (3, 4), w = (3, 19)

Finding an Image and a Preimage In Exercises 18, use the function to find (a) the image of v and (b) the preimage of w. T(v1, v2) = (v1, 2v2 v1, v2), v = (0, 4), w = (2, 4, 3)

Finding an Image and a Preimage In Exercises 18, use the function to find (a) the image of v and (b) the preimage of w. T(v1, v2, v3) = (2v1 + v2, 2v2 3v1, v1 v3), v = (4, 5, 1), w = (4, 1, 1)

Finding an Image and a Preimage In Exercises 18, use the function to find (a) the image of v and (b) the preimage of w. T(v1, v2, v3) = (v2 v1, v1 + v2, 2v1), v = (2, 3, 0), w = (11, 1, 10)

Finding an Image and a Preimage In Exercises 18, use the function to find (a) the image of v and (b) the preimage of w. T(v1, v2, v3) = (4v2 v1, 4v1 + 5v2), v = (2, 3, 1), w = (3, 9)

Finding an Image and a Preimage In Exercises 18, use the function to find (a) the image of v and (b) the preimage of w. T(v1, v2, v3) = (2v1 + v2, v1 v2), v = (2, 1, 4), w = (1, 2)

Finding an Image and a Preimage In Exercises 18, use the function to find (a) the image of v and (b) the preimage of w. T(v1, v2) = ( 2 2 v1 2 2 v2, v1 + v2, 2v1 v2), v = (1, 1), w = (52, 2, 16)

Finding an Image and a Preimage In Exercises 18, use the function to find (a) the image of v and (b) the preimage of w. T(v1, v2) = ( 3 2 v1 1 2 v2, v1 v2, v2), v = (2, 4), w = (3, 2, 0)

Linear Transformations In Exercises 922, determine whether the function is a linear transformation. T: R2R2, T(x, y) = (x, 1)

Linear Transformations In Exercises 922, determine whether the function is a linear transformation. T: R2R2, T(x, y) = (x, y2)

Linear Transformations In Exercises 922, determine whether the function is a linear transformation. T: R3R3, T(x, y, z) = (x + y, x y, z)

Linear Transformations In Exercises 922, determine whether the function is a linear transformation. T: R3R3, T(x, y, z) = (x + 1, y + 1, z + 1)

Linear Transformations In Exercises 922, determine whether the function is a linear transformation. T: R2R3, T(x, y) = (x, xy, y)

Linear Transformations In Exercises 922, determine whether the function is a linear transformation. T: R2R3, T(x, y) = (x2, xy, y2)

Linear Transformations In Exercises 922, determine whether the function is a linear transformation. T: M2,2R, T(A) = A

Linear Transformations In Exercises 922, determine whether the function is a linear transformation. T: M2,2R, T(A) = a + b + c + d, where A = [ a c b d].

Linear Transformations In Exercises 922, determine whether the function is a linear transformation. T: M2,2R, T(A) = a b c d, where A = [ a c b d].

Linear Transformations In Exercises 922, determine whether the function is a linear transformation. T: M2,2R, T(A) = b2, where A = [ a c b d].

Linear Transformations In Exercises 922, determine whether the function is a linear transformation. T: M3,3M3,3, T(A) = [ 0 0 1 0 1 0 1 0 0 ]A

Linear Transformations In Exercises 922, determine whether the function is a linear transformation. T: M3,3M3,3, T(A) = [ 3 0 0 0 2 0 0 0 10]A

Linear Transformations In Exercises 922, determine whether the function is a linear transformation. T: P2P2, T(a0 + a1x + a2x2) = (a0 + a1 + a2) + (a1 + a2)x + a2x2

Linear Transformations In Exercises 922, determine whether the function is a linear transformation. T: P2P2, T(a0 + a1x + a2x2) = a1 + 2a2x

Let T be a linear transformation from R2 into R2 such that T(1, 0) = (1, 1) and T(0, 1) = (1, 1). Find T(1, 4) and T(2, 1).

Let T be a linear transformation from R2 into R2 such that T(1, 2) = (1, 0) and T(1, 1) = (0, 1). Find T(2, 0) and T(0, 3).

Linear Transformation and Bases In Exercises 2528, let T: R3R3 be a linear transformation such that T(1, 0, 0) = (2, 4, 1), T(0, 1, 0) = (1, 3, 2), and T(0, 0, 1) = (0, 2, 2). Find the specified image. T(1, 3, 0)

Linear Transformation and Bases In Exercises 2528, let T: R3R3 be a linear transformation such that T(1, 0, 0) = (2, 4, 1), T(0, 1, 0) = (1, 3, 2), and T(0, 0, 1) = (0, 2, 2). Find the specified image. T(2, 1, 0)

Linear Transformation and Bases In Exercises 2528, let T: R3R3 be a linear transformation such that T(1, 0, 0) = (2, 4, 1), T(0, 1, 0) = (1, 3, 2), and T(0, 0, 1) = (0, 2, 2). Find the specified image. T(2, 4, 1)

Linear Transformation and Bases In Exercises 2528, let T: R3R3 be a linear transformation such that T(1, 0, 0) = (2, 4, 1), T(0, 1, 0) = (1, 3, 2), and T(0, 0, 1) = (0, 2, 2). Find the specified image. T(2, 4, 1)

Linear Transformation and Bases In Exercises 2932, let T: R3R3 be a linear transformation such that T(1, 1, 1) = (2, 0, 1), T(0, 1, 2) = (3, 2, 1), and T(1, 0, 1) = (1, 1, 0). Find the specified image. T(4, 2, 0)

Linear Transformation and Bases In Exercises 2932, let T: R3R3 be a linear transformation such that T(1, 1, 1) = (2, 0, 1), T(0, 1, 2) = (3, 2, 1), and T(1, 0, 1) = (1, 1, 0). Find the specified image. T(0, 2, 1)

Linear Transformation and Bases In Exercises 2932, let T: R3R3 be a linear transformation such that T(1, 1, 1) = (2, 0, 1), T(0, 1, 2) = (3, 2, 1), and T(1, 0, 1) = (1, 1, 0). Find the specified image. T(2, 1, 1)

Linear Transformation and Bases In Exercises 2932, let T: R3R3 be a linear transformation such that T(1, 1, 1) = (2, 0, 1), T(0, 1, 2) = (3, 2, 1), and T(1, 0, 1) = (1, 1, 0). Find the specified image. T(2, 1, 0)

Linear Transformation Given by a Matrix In Exercises 3338, define the linear transformation T: RnRm by T(v) = Av. Find the dimensions of Rn and Rm. A = [ 0 1 1 0]

Linear Transformation Given by a Matrix In Exercises 3338, define the linear transformation T: RnRm by T(v) = Av. Find the dimensions of Rn and Rm. A = [ 1 2 2 2 4 2 ]

Linear Transformation Given by a Matrix In Exercises 3338, define the linear transformation T: RnRm by T(v) = Av. Find the dimensions of Rn and Rm. A = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 2 ]

Linear Transformation Given by a Matrix In Exercises 3338, define the linear transformation T: RnRm by T(v) = Av. Find the dimensions of Rn and Rm. A = [ 1 0 2 0 1 2 3 1 4 0]

Linear Transformation Given by a Matrix In Exercises 3338, define the linear transformation T: RnRm by T(v) = Av. Find the dimensions of Rn and Rm. A = [ 0 1 0 1 4 1 2 5 3 1 0 1 ]

Linear Transformation Given by a Matrix In Exercises 3338, define the linear transformation T: RnRm by T(v) = Av. Find the dimensions of Rn and Rm. . A = [ 0 1 1 2 0 2 0 1 2 2 0 2 0 1 1 ]

For the linear transformation from Exercise 33, find (a) T(1, 1), (b) the preimage of (1, 1), and (c) the preimage of (0, 0).

Writing For the linear transformation from Exercise 34, find (a) T(2, 4) and (b) the preimage of (1, 2, 2). (c) Then explain why the vector (1, 1, 1) has no preimage under this transformation.

For the linear transformation from Exercise 35, find (a) T(2, 1, 2, 1) and (b) the preimage of (1, 1, 1, 1).

For the linear transformation from Exercise 36, find (a) T(1, 0, 1, 3, 0) and (b) the preimage of (1, 8).

For the linear transformation from Exercise 37, find (a) T(1, 0, 2, 3) and (b) the preimage of (0, 0, 0).

For the linear transformation from Exercise 38, find (a) T(0, 1, 0, 1, 0) (b) the preimage of (0, 0, 0), and (c) the preimage of (1, 1, 2).

Let T be a linear transformation from R2 into R2 such that T(x, y) = (x cos y sin , x sin + y cos ). Find (a) T(4, 4) for = 45, (b) T(4, 4) for = 30, and (c) T(5, 0) for = 120.

For the linear transformation from Exercise 45, let = 45 and find the preimage of v = (1, 1).

Find the inverse of the matrix A in Example 7. What linear transformation from R2 into R2 does A1 represent?

For the linear transformation T: R2R2 given by A = [ a b b a] find a and b such that T(12, 5) = (13, 0).

Projection in R3 In Exercises 49 and 50, let the matrix A represent the linear transformation T: R3R3. Describe the orthogonal projection to which T maps every vector in R3. A = [ 1 0 0 0 0 0 0 0 1 ]

Projection in R3 In Exercises 49 and 50, let the matrix A represent the linear transformation T: R3R3. Describe the orthogonal projection to which T maps every vector in R3. A = [ 0 0 0 0 1 0 0 0 1 ]

Linear Transformation Given by a Matrix In Exercises 5154, determine whether the function involving the n n matrix A is a linear transformation. T: Mn,nMn,n, T(A) = A1

Linear Transformation Given by a Matrix In Exercises 5154, determine whether the function involving the n n matrix A is a linear transformation. T: Mn,nMn,n, T(A) = AX XA, where X is a fixed n m matrix

Linear Transformation Given by a Matrix In Exercises 5154, determine whether the function involving the n n matrix A is a linear transformation. T: Mn,nMn,m, T(A) = AB, where B is a fixed n m matrix

Linear Transformation Given by a Matrix In Exercises 5154, determine whether the function involving the n n matrix A is a linear transformation. T: Mn,nR, T(A) = a11 a22 . . . ann, where A = [aij]

Let T be a linear transformation from P2 into P2 such that T(1) = x, T(x) = 1 + x, and T(x2) = 1 + x + x2. Find T(2 6x + x2).

Let T be a linear transformation from M2,2 into M2,2 such that T([ 1 0 0 0]) = [ 1 0 1 2], T([ 0 0 1 0]) = [ 0 1 2 1], T([ 0 1 0 0]) = [ 1 0 2 1], T([ 0 0 0 1]) = [ 3 1 1 0]. Find T([ 1 1 3 4]).

Calculus In Exercises 5760, let Dx be the linear transformation from C[a, b] into C[a, b] from Example 10. Determine whether each statement is true or false. Explain. Dx(ex2 + 2x) = Dx(ex2 ) + 2Dx(x)

Calculus In Exercises 5760, let Dx be the linear transformation from C[a, b] into C[a, b] from Example 10. Determine whether each statement is true or false. Explain. Dx(x2 ln x) = Dx(x2) Dx(ln x)

Calculus In Exercises 5760, let Dx be the linear transformation from C[a, b] into C[a, b] from Example 10. Determine whether each statement is true or false. Explain. Dx(sin 3x) = 3Dx(sin x)

Calculus In Exercises 5760, let Dx be the linear transformation from C[a, b] into C[a, b] from Example 10. Determine whether each statement is true or false. Explain. Dx(cos x 2) = 1 2 Dx(cos x)

Calculus In Exercises 6164, for the linear transformation from Example 10, find the preimage of each function. Dx( f ) = 4x + 3

Calculus In Exercises 6164, for the linear transformation from Example 10, find the preimage of each function. Dx( f ) = ex

Calculus In Exercises 6164, for the linear transformation from Example 10, find the preimage of each function. Dx( f ) = sin x

Calculus In Exercises 6164, for the linear transformation from Example 10, find the preimage of each function. Dx( f ) = 1 x

Calculus Let T be a linear transformation from P into R such that T(p) = 1 0 p(x) dx. Find (a) T(2 + 3x2), (b) T(x3 x5), and (c) T(6 + 4x).

Calculus Let T be the linear transformation from P2 into R using the integral in Exercise 65. Find the preimage of 1. That is, find the polynomial function(s) of degree 2 or less such that T(p) = 1.

True or False? In Exercises 67 and 68, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The function f(x) = cos x is a linear transformation from R into R. (b) For polynomials, the differential operator Dx is a linear transformation from Pn into Pn1.

True or False? In Exercises 67 and 68, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The function g(x) = x3 is a linear transformation from R into R. (b) Any linear function of the form f(x) = ax + b is a linear transformation from R into R.

Writing Let T: R2R2 such that T(1, 0) = (1, 0) and T(0, 1) = (0, 0). (a) Determine T(x, y) for (x, y) in R2. (b) Give a geometric description of T.

Writing Let T: R2R2 such that T(1, 0) = (0, 1) and T(0, 1) = (1, 0). (a) Determine T(x, y) for (x, y) in R2. (b) Give a geometric description of T.

Proof Let T be the function that maps R2 into R2 such that T(u) = projvu, where v = (1, 1). (a) Find T(x, y). (b) Find T(5, 0). (c) Prove that T is a linear transformation from R2 into R2

Writing Find T(3, 4) and T(T(3, 4)) from Exercise 71 and give geometric descriptions of the results.

Show that T from Exercise 71 is represented by the matrix A = [ 1 2 1 2 1 2 1 2 ].

CAPSTONE Explain how to determine whether a function T: VW is a linear transformation.

Proof Use the concept of a fixed point of a linear transformation T: VV. A vector u is a fixed point when T(u) = u. (a) Prove that 0 is a fixed point of any linear transformation T: VV. (b) Prove that the set of fixed points of a linear transformation T: VV is a subspace of V. (c) Determine all fixed points of the linear transformation T: R2R2 represented by T(x, y) = (x, 2y). (d) Determine all fixed points of the linear transformation T: R2R2 represented by T(x, y) = (y, x).

A translation in R2 is a function of the form T(x, y) = (x h, y k), where at least one of the constants h and k is nonzero. (a) Show that a translation in R2 is not a linear transformation. (b) For the translation T(x, y) = (x 2, y + 1), determine the images of (0, 0), (2, 1), and (5, 4). (c) Show that a translation in R2 has no fixed points.

Proof Prove that (a) the zero transformation and (b) the identity transformation are linear transformations.

Let S = {v1, v2, v3} be a set of linearly independent vectors in R3. Find a linear transformation T from R3 into R3 such that the set {T(v1), T(v2), T(v3)} is linearly dependent.

Proof Let S = {v1, v2, . . . , vn} be a set of linearly dependent vectors in V, and let T be a linear transformation from V into V. Prove that the set {T(v1), T(v2), . . . , T(vn)} is linearly dependent.

Proof Let V be an inner product space. For a fixed vector v0 in V, define T: VR by T(v) = v, v0. Prove that T is a linear transformation.

Proof Define T: Mn,nR by T(A) = a11 + a22 + . . . + ann (the trace of A). Prove that T is a linear transformation.

Let V be an inner product space with a subspace W having B = {w1, w2, . . . , wn} as an orthonormal basis. Show that the function T: VW represented by T(v) = v, w1w1 + v, w2w2 + . . . + v, wnwn is a linear transformation. T is called the orthogonal projection of V onto W.

Guided Proof Let {v1, v2, . . . , vn} be a basis for a vector space V. Prove that if a linear transformation T: VV satisfies T(vi ) = 0 for i = 1, 2, . . . , n, then T is the zero transformation. Getting Started: To prove that T is the zero transformation, you need to show that T(v) = 0 for every vector v in V. (i) Let v be an arbitrary vector in V such that v = c1v1 + c2v2 + . . . + cnvn. (ii) Use the definition and properties of linear transformations to rewrite T(v) as a linear combination of T(vi ). (iii) Use the fact that T(vi ) = 0 to conclude that T(v) = 0, making T the zero transformation.

Guided Proof Prove that T: VW is a linear transformation if and only if T(au + bv) = aT(u) + bT(v) for all vectors u and v and all scalars a and b. Getting Started: This is an if and only if statement, so you need to prove the statement in both directions. To prove that T is a linear transformation, you need to show that the function satisfies the definition of a linear transformation. In the other direction, let T be a linear transformation. Use the definition and properties of a linear transformation to prove that T(au + bv) = aT(u) + bT(v). (i) Let T(au + bv) = aT(u) + bT(v). Show that T preserves the properties of vector addition and scalar multiplication by choosing appropriate values of a and b. (ii) To prove the statement in the other direction, assume that T is a linear transformation. Use the properties and definition of a linear transformation to show that T(au + bv) = aT(u) + bT(v)..

In Exercises 1–8, use the function to find (a) the image of v and (b) the preimage of w. \(\begin{array}{l} T\left(v_{1}, v_{2}\right)=\left(v_{1}+v_{2}, v_{1}-v_{2}\right), \\ \mathbf{v}=(3,-4), \mathbf{w}=(3,19) \end{array} \) Text Transcription: T(v_1, v_2) = (v_1 + v_2, v_1 - v_2), v = (3, -4), w = (3, 19)

In Exercises 1–8, use the function to find (a) the image of v and (b) the preimage of w. \(T(v_{1}, v_{2}) = (v_{1}, 2v_{2} ? v_{1}, v_{2})\), v = (0, 4), w = (2, 4, 3) Text Transcription: T(v_1, v_2) = (v_1, 2v_2 ? v_1, v_2)

In Exercises 1–8, use the function to find (a) the image of v and (b) the preimage of w. \(T(v_{1}, v_{2}, v_{3}) = (2v_{1} + v_{2}, 2v_{2} ? 3v_{1}, v_{1} ? v_{3})\), v = (?4, 5, 1), w = (4, 1, ?1) Text Transcription: T(v_1, v_2, v_3) = (2v_1 + v_2, 2v_2 ? 3v_1, v_1 ? v_3)

In Exercises 1–8, use the function to find (a) the image of v and (b) the preimage of w. \(T(v_{1}, v_{2}, v_{3}) = (v_{2} ? v_{1}, v_{1} + v_{2}, 2v_{1})\), v = (2, 3, 0), w = (?11, ?1, 10) Text Transcription: T(v_1, v_2, v_3) = (v_2 ? v_1, v_1 + v_2, 2v_1)

In Exercises 1–8, use the function to find (a) the image of v and (b) the preimage of w. \(T(v_{1}, v_{2}, v_{3}) = (4v_{2} ? v_{1}, 4v_{1} + 5v_{2})\), v = (2, ?3, ?1), w = (3, 9) Text Transcription: T(v_1, v_2, v_3) = (4v_2 ? v_1, 4v_1 + 5v_2)

In Exercises 1–8, use the function to find (a) the image of v and (b) the preimage of w. \(T(v_{1}, v_{2}, v_{3}) = (2v_{1} + v_{2}, v_{1} ? v_{2})\), v = (2, 1, 4), w = (?1, 2) Text Transcription: T(v_1, v_2, v_3) = (2v_1 + v_2, v_1 ? v_2)

In Exercises 1–8, use the function to find (a) the image of v and (b) the preimage of w. \(\begin{array}{l} T\left(v_{1}, v_{2}\right)=\left(\frac{\sqrt{2}}{2} v_{1}-\frac{\sqrt{2}}{2} v_{2}, v_{1}+v_{2}, 2 v_{1}-v_{2}\right) \\ \mathbf{v}=(1,1), \mathbf{w}=(-5 \sqrt{2},-2,-16) \end{array} \) Text Transcription: T(v_1, v_2) = ( sqrt 2 / 2 v_1 - sqrt 2 / 2 v_2, v_1 + v_2, 2v_1 - v_2) v = (1,1), w = (-5 sqrt 2, -2, -16)

In Exercises 1–8, use the function to find (a) the image of v and (b) the preimage of w. \(\begin{array}{l} T\left(v_{1}, v_{2}\right)=\left(\frac{\sqrt{3}}{2} v_{1}-\frac{1}{2} v_{2}, v_{1}-v_{2}, v_{2}\right) \\ \mathbf{v}=(2,4), \mathbf{w}=(\sqrt{3}, 2,0) \end{array} \) Text Transcription: T(v_1, v_2) = ( sqrt 3/2 v_1 - 1/2 v_2, v_1 - v_2, v_2) v = (2, 4), w = (sqrt 3, 2, 0)

In Exercises 9–22, determine whether the function is a linear transformation. \(T: R^{2} \rightarrow R^{2}, T(x, y)=(x, 1)\) Text Transcription: T: R^2 rightarrow R^2, T(x, y) = (x, 1)

In Exercises 9–22, determine whether the function is a linear transformation. \(T: R^{2} \rightarrow R^{2}, T(x, y) = (x, y^{2})\) Text Transcription: T: R^2 rightarrow R^2, T(x, y) = (x, y^2)

In Exercises 9–22, determine whether the function is a linear transformation. \(T: R^{3} \rightarrow R^{3}, T(x, y, z) = (x + y, x ? y, z)\) Text Transcription: T: R^3 rightarrow R^3, T(x, y, z) = (x + y, x ? y, z)

In Exercises 9–22, determine whether the function is a linear transformation. \(T: R^{3} \rightarrow R^{3}, T(x, y, z) = (x + 1, y + 1, z + 1)\) Text Transcription: T: R^3 righarrow R^3, T(x, y, z) = (x + 1, y + 1, z + 1)

In Exercises 9–22, determine whether the function is a linear transformation. \(T: R^{2} \rightarrow R^{3}, T(x, y)=(\sqrt{x}, x y, \sqrt{y})\) Text Transcription: T: R^2 rightarrow R^3, T(x, y) = (sqrt x, xy, sqrt y)

In Exercises 9–22, determine whether the function is a linear transformation. \(T: R^{2} \rightarrow R^{3}, T(x, y) = (x^{2}, xy, y^{2})\) Text Transcription: T: R^2 rightarrow R^3, T(x, y) = (x^2, xy, y^2)

In Exercises 9–22, determine whether the function is a linear transformation. \(T: M_{2,2} \rightarrow R, T(A)=|A|\) Text Transcription: T: M_2,2 rightarrow R, T(A) = |A|

In Exercises 9–22, determine whether the function is a linear transformation. \(T: M_{2.2} \rightarrow R, T(A)=a+b+c+d\), where \(A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \). Text Transcription: T: M_2,2 rightarrow R, T(A) = a + b + c + d A = [ _c^a _d^b]

In Exercises 9–22, determine whether the function is a linear transformation. \(T: M_{2.2} \rightarrow R, T(A)=a-b-c-d\), where \(A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \). Text Transcription: T: M_2,2 rightarrow R, T(A) = a - b - c - d A = [ _c^a _d^b]

In Exercises 9–22, determine whether the function is a linear transformation. \(T: M_{2,2} \rightarrow R, T(A)=b^{2}\), where \(A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \). Text Transcription: T: M_2,2 rightarrow R, T(A) = b^2 A = [ _c^a _d^b]

In Exercises 9–22, determine whether the function is a linear transformation. \(T: M_{3,3} \rightarrow M_{3,3}, T(A)=\left[\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right] A \), Text Transcription: T: M_3,3 rightarrow M_3,3, T(A) = [_1^0^0 _0^1^0 _0^0^1] A

In Exercises 9–22, determine whether the function is a linear transformation. \(T: M_{3,3} \rightarrow M_{3,3}, T(A)=\left[\begin{array}{rrr} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -10 \end{array}\right] A \), Text Transcription: T: M_3,3 rightarrow M_3,3, T(A) = [_0^0^3 _0^2^0 _-10^0^0] A

In Exercises 9–22, determine whether the function is a linear transformation. \(\begin{array}{l} T: P_{2} \rightarrow P_{2}, T\left(a_{0}+a_{1} x+a_{2} x^{2}\right)= \\ \left(a_{0}+a_{1}+a_{2}\right)+\left(a_{1}+a_{2}\right) x+a_{2} x^{2} \end{array} \), Text Transcription: T: P_2P_2, T(a_0 + a_1x + a_2x^2) = (a_0 + a_1 + a_2) + (a_1 + a_2)x + a_2x^2

In Exercises 9–22, determine whether the function is a linear transformation. \(T: P_{2} \rightarrow P_{2}, T\left(a_{0}+a_{1} x+a_{2} x^{2}\right)=a_{1}+2 a_{2} x\), Text Transcription: T: P_2 rightarrow P_2, T(a_0 + a_1x + a_2x^2) = a_1 + 2a_2x

Let T be a linear transformation from \(R^{2}\) into \(R^{2}\) such that T(1, 0) = (1, 1) and T(0, 1) = (?1, 1). Find T(1, 4) and T(?2, 1). Text Transcription: R^2

Let T be a linear transformation from \(R^{2}\) into \(R^{2}\) such that T(1, 2) = (1, 0) and T(?1, 1) = (0, 1). Find T(2, 0) and T(0, 3). Text Transcription: R^2

In Exercises 25–28, let \(T: R^{3} \rightarrow R^{3}\) be a linear transformation such that T1, 0, 0 2, 4, 1, T0, 1, 0 1, 3, 2, and T0, 0, 1 0, 2, 2. Find the specified image. T(1, ?3, 0) Text Transcription: T: R^3 rightarrow R^3

In Exercises 25–28, let \(T: R^{3} \rightarrow R^{3}\) be a linear transformation such that T1, 0, 0 2, 4, 1, T0, 1, 0 1, 3, 2, and T0, 0, 1 0, 2, 2. Find the specified image. T(2, ?1, 0) Text Transcription: T: R^3 rightarrow R^3

In Exercises 25–28, let \(T: R^{3} \rightarrow R^{3}\) be a linear transformation such that T1, 0, 0 2, 4, 1, T0, 1, 0 1, 3, 2, and T0, 0, 1 0, 2, 2. Find the specified image. T(2, ?4, 1) Text Transcription: T: R^3 rightarrow R^3

In Exercises 25–28, let \(T: R^{3} \rightarrow R^{3}\) be a linear transformation such that T1, 0, 0 2, 4, 1, T0, 1, 0 1, 3, 2, and T0, 0, 1 0, 2, 2. Find the specified image. T(?2, 4, ?1) Text Transcription: T: R^3 rightarrow R^3

In Exercises 29–32, let T: R3R3 be a linear transformation such that T1, 1, 1 2, 0, 1, T0, 1, 2 3, 2, 1, and T1, 0, 1 1, 1, 0. Find the specified image. T(4, 2, 0) Text Transcription: T: R^3 rightarrow R^3

In Exercises 29–32, let T: R3R3 be a linear transformation such that T1, 1, 1 2, 0, 1, T0, 1, 2 3, 2, 1, and T1, 0, 1 1, 1, 0. Find the specified image. T(0, 2, ?1) Text Transcription: T: R^3 rightarrow R^3

In Exercises 29–32, let T: R3R3 be a linear transformation such that T1, 1, 1 2, 0, 1, T0, 1, 2 3, 2, 1, and T1, 0, 1 1, 1, 0. Find the specified image. T(2, ?1, 1) Text Transcription: T: R^3 rightarrow R^3

In Exercises 29–32, let T: R3R3 be a linear transformation such that T1, 1, 1 2, 0, 1, T0, 1, 2 3, 2, 1, and T1, 0, 1 1, 1, 0. Find the specified image. T(?2, 1, 0) Text Transcription: T: R^3 rightarrow R^3

In Exercises 33–38, define the linear transformation \(T: R^{n} \rightarrow R^{m}\) by Tv = Av. Find the dimensions of \(R^{n}\) and \(R^{m}\). \(A=\left[\begin{array}{rr} 0 & -1 \\ -1 & 0 \end{array}\right] \) Text Transcription: T: R^n rightarrow R^m R^n R^m A = [_-1^0 _0^-1]

In Exercises 33–38, define the linear transformation \(T: R^{n} \rightarrow R^{m}\) by Tv = Av. Find the dimensions of \(R^{n}\) and \(R^{m}\). \(A=\left[\begin{array}{rr} 1 & 2 \\ -2 & 4 \\ -2 & 2 \end{array}\right] \) Text Transcription: T: R^n rightarrow R^m R^n R^m A = [_-2^-2^1 _2^4^2]

In Exercises 33–38, define the linear transformation \(T: R^{n} \rightarrow R^{m}\) by Tv = Av. Find the dimensions of \(R^{n}\) and \(R^{m}\). \(A=\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right] \) Text Transcription: T: R^n rightarrow R^m R^n R^m A = [_0^0^0^1 _0^0^-1^0 _0^1^0^0 _2^0^0^0]

In Exercises 33–38, define the linear transformation \(T: R^{n} \rightarrow R^{m}\) by Tv = Av. Find the dimensions of \(R^{n}\) and \(R^{m}\). \(A=\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right] \) Text Transcription: T: R^n rightarrow R^m R^n R^m A = [_0^-1 _0^2 _2^1 _-1^3 _0^4]

In Exercises 33–38, define the linear transformation \(T: R^{n} \rightarrow R^{m}\) by Tv = Av. Find the dimensions of \(R^{n}\) and \(R^{m}\). \(A=\left[\begin{array}{rrrr} 0 & 1 & -2 & 1 \\ -1 & 4 & 5 & 0 \\ 0 & 1 & 3 & 1 \end{array}\right] \) Text Transcription: T: R^n rightarrow R^m R^n R^m A = [_0^-1^0 _1^4^1 _3^5^-2 _1^0^1]

Linear Transformation Given by a Matrix In Exercises 3338, define the linear transformation T: RnRm by T(v) = Av. Find the dimensions of Rn and Rm. . A = [ 0 1 1 2 0 2 0 1 2 2 0 2 0 1 1 ]

For the linear transformation from Exercise 33, find (a) T(1, 1), (b) the preimage of (1, 1), and (c) the preimage of (0, 0).

For the linear transformation from Exercise 34, find (a) T(2, 4) and (b) the preimage of (?1, 2, 2). (c) Then explain why the vector (1, 1, 1) has no preimage under this transformation.

For the linear transformation from Exercise 35, find (a) T(2, 1, 2, 1) and (b) the preimage of (?1, ?1, ?1, ?1).

For the linear transformation from Exercise 36, find (a) T(1, 0, ?1, 3, 0) and (b) the preimage of (?1, 8).

For the linear transformation from Exercise 37, find (a) T(1, 0, 2, 3) and (b) the preimage of (0, 0, 0).

For the linear transformation from Exercise 38, find (a) T(0, 1, 0, 1, 0) (b) the preimage of (0, 0, 0), and (c) the preimage of (1, ?1, 2).

Let T be a linear transformation from \(R^{2}\) into \(R^{2}\) such that \(T(x, y) = (x cos \theta ? y sin \theta, x sin \theta + y cos \theta)\). Find (a) T(4, 4) for \(\theta = 45°\), (b) T(4, 4) for \(\theta = 30°\), and (c) T(5, 0) for \(\theta = 120°\). Text Transcription: R^2 T(x, y) = (x cos theta ? y sin theta , x sin theta + y cos theta) theta = 45° theta = 30° theta = 120°

For the linear transformation from Exercise 45, let \(\theta = 45°\) and find the preimage of v = (1, 1). Text Transcription: theta = 45°

Find the inverse of the matrix A in Example 7. What linear transformation from \(R^{2}\) into \(R^{2}\) does \(A^{?1}\) represent? Text Transcription: R^2 A^-1

For the linear transformation \(T: R^{2} \rightarrow R^{2}\) given by \(A=\left[\begin{array}{rr} a & -b \\ b & a \end{array}\right] \) find a and b such that T(12, 5) = (13, 0). Text Transcription: T: R^2 rightarrow R^2 A = [_b^a _a^-b]

In Exercises 49 and 50, let the matrix A represent the linear transformation \(T: R^{3} \rightarrow R^{3}\). Describe the orthogonal projection to which T maps every vector in \(R^{3}\). \(A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right] \) Text Transcription: T: R^3 rightarrow R^3 R^3 A = [_0^01 _0^0^0 _1^0^0]

In Exercises 49 and 50, let the matrix A represent the linear transformation \(T: R^{3} \rightarrow R^{3}\). Describe the orthogonal projection to which T maps every vector in \(R^{3}\). \(A=\left[\begin{array}{lll} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \) Text Transcription: T: R^3 rightarrow R^3 R^3 A = [_0^0^0 _0^1^0 _1^0^0]

In Exercises 51–54, determine whether the function involving the n x n matrix A is a linear transformation. \(T: M_{n, n} \rightarrow M_{n, n}, T(A)=A^{-1}\) Text Transcription: T: M_n, n rightarrow M_n, n, T(A)=A^-1

In Exercises 51–54, determine whether the function involving the n x n matrix A is a linear transformation. \(T: M_{n, n} \rightarrow M_{n, n}, T(A)=A X-X A\), where X is a fixed n × m matrix Text Transcription: T: M_n, n rightarrow M_n, n, T(A)=A X-X A

In Exercises 51–54, determine whether the function involving the n x n matrix A is a linear transformation. \(T: M_{n, n} \rightarrow M_{n, n}, T(A)=AB\), where B is a fixed n × m matrix Text Transcription: T: M_n, n rightarrow M_n, n, T(A)=AB

In Exercises 51–54, determine whether the function involving the n x n matrix A is a linear transformation. \(T: M_{n, n} \rightarrow R, T(A)=a_{11} \cdot a_{22} \cdots \cdots \cdot a_{n n} \text {, where } A=\left[a_{i j}\right]\), Text Transcription: T: M_n, n rightarrow R, T(A)=a_11 cdot a_22 cdots cdots cdot a_{n n} where [a_i j]

Let T be a linear transformation from \(P^{2}\) into \(P^{2}\) such that T(1) = x, T(x) = 1 + x, and \(T(x^{2}) = 1 + x + x^{2}\). Find \(T(2 ? 6x + x^{2})\). Text Transcription: P_2 T(x^2) = 1 + x + x^2 T(2 - 6x + x^2)

Let T be a linear transformation from \(M_{2,2}\) into \(M_{2,2}\) such that \(\begin{array}{l} T\left(\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right]\right)=\left[\begin{array}{rr} 1 & -1 \\ 0 & 2 \end{array}\right], \quad T\left(\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right]\right)=\left[\begin{array}{ll} 0 & 2 \\ 1 & 1 \end{array}\right], \\ T\left(\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right]\right)=\left[\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}\right], \quad T\left(\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right]\right)=\left[\begin{array}{rr} 3 & -1 \\ 1 & 0 \end{array}\right] . \\ \text { Find } T\left(\left[\begin{array}{rr} 1 & 3 \\ -1 & \end{array}\right]\right) . \end{array} \) Text Transcription: T([_0^1 _0^1]) = [_0^1 _2^-1], T ([_0^0 _0^1]) = [_1^0 _1^2], T_([_1^0 _0^0]) = [_0^1 _1^2], T([_0^0 _1^0]) = [_1^3 _0^-1]. Find T[(_-1^1 _4^3]).

In Exercises 57–60, let \(D_{x}\) be the linear transformation from C’[a, b] into C’[a, b] from Example 10. Determine whether each statement is true or false. Explain. \(D_{x}\left(e^{x^{2}}+2 x\right)=D_{x}\left(e^{x^{2}}\right)+2 D_{x}(x)\) Text Transcription: D_x D_x(e^x^2 + 2x) = D_x(e^x^2) + 2D_x(x)

In Exercises 57–60, let \(D_{x}\) be the linear transformation from C’[a, b] into C’[a, b] from Example 10. Determine whether each statement is true or false. Explain. \(D_{x}\left(x^{2}-\ln x\right)=D_{x}\left(x^{2}\right)-D_{x}(\ln x)\) Text Transcription: D_x D_x(x^2 ? ln x) = D_x(x^2) ? D_x(ln x)

In Exercises 57–60, let \(D_{x}\) be the linear transformation from C’[a, b] into C’[a, b] from Example 10. Determine whether each statement is true or false. Explain. \(D_{x}(\sin 3 x)=3 D_{x}(\sin x)\) Text Transcription: D_x D_x(sin 3x) = 3D_x(sin x)

In Exercises 57–60, let \(D_{x}\) be the linear transformation from C’[a, b] into C’[a, b] from Example 10. Determine whether each statement is true or false. Explain. \(D_{x}\left(\cos \frac{x}{2}\right)=\frac{1}{2} D_{x}(\cos x)\) Text Transcription: D_x D_x(cos x/2) = 1/2 D_x (cos x)

In Exercises 61–64, for the linear transformation from Example 10, find the preimage of each function. \(D_{x}( f ) = 4x + 3\) Text Transcription: D_x( f ) = 4x + 3

In Exercises 61–64, for the linear transformation from Example 10, find the preimage of each function. \(D_{x}( f ) = e^{x}\) Text Transcription: D_x( f ) = e^x

In Exercises 61–64, for the linear transformation from Example 10, find the preimage of each function. \(D_{x}( f ) = sin x\) Text Transcription: D_x( f ) = sin x

In Exercises 61–64, for the linear transformation from Example 10, find the preimage of each function. \(D_{x}(f)=\frac{1}{x}\) Text Transcription: D_x( f ) = 1/x

Let T be a linear transformation from P into R such that \(T(p)=\int_{0}^{1} p(x) d x\). Find (a) \(T(?2 + 3x^{2})\), (b) \(T(x^{3}?x^{5})\), and (c) T(?6+4x). Text Transcription: T(p) = integ^1_0 p(x) dx T(-2 + 3x^2) T(x^3 - x^5)

Let T be the linear transformation from \(P_{2}\) into R using the integral in Exercise 65. Find the preimage of 1. That is, find the polynomial function(s) of degree 2 or less such that T(p) = 1. Text Transcription: P_2

In Exercises 67 and 68, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The function f (x) = cos x is a linear transformation from R into R. (b) For polynomials, the differential operator \(D_{x}\) is a linear transformation from \(P_{n}\) into \(P_{n?1}\). Text Transcription: D_x P_n P_n-1

In Exercises 67 and 68, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The function \(g(x) = x^{3}\) is a linear transformation from R into R. (b) Any linear function of the form f (x) = ax + b is a linear transformation from R into R. Text Transcription: g(x) = x^3

Let \(T: R^{2} \rightarrow R^{2}\) such that T(1, 0) = (1, 0) and T(0, 1) = (0, 0). (a) Determine T(x, y) for (x, y) in \(R^{2}\). (b) Give a geometric description of T. Text Transcription: T: R^2 rightarrow R^2 R^2

Let \(T: R^{2} \rightarrow R^{2}\) such that T(1, 0) = (0, 1) and T(0, 1) = (1, 0). (a) Determine T(x, y) for (x, y) in \(R^{2}\). (b) Give a geometric description of T. Text Transcription: T: R^2 rightarrow R^2 R^2

Proof Let T be the function that maps R2 into R2 such that T(u) = projvu, where v = (1, 1). (a) Find T(x, y). (b) Find T(5, 0). (c) Prove that T is a linear transformation from R2 into R2

Find T(3, 4) and T(T(3, 4)) from Exercise 71 and give geometric descriptions of the results.

Show that T from Exercise 71 is represented by the matrix \(A=\left[\begin{array}{ll} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{array}\right]\) Text Transcription: A = [_1/2 1/2 ^1/2 1/2]

Explain how to determine whether a function \(T: V \rightarrow W\) is a linear transformation. Text Transcription: T: V rightarrow W

Use the concept of a fixed point of a linear transformation \(T: V \rightarrow V\). A vector u is a fixed point when T(u) = u. (a) Prove that 0 is a fixed point of any linear transformation \(T: V \rightarrow V\). (b) Prove that the set of fixed points of a linear transformation \(T: V \rightarrow V\) is a subspace of V. (c) Determine all fixed points of the linear transformation \(T: R^{2} \rightarrow R^{2}\) represented by T(x, y) = (x, 2y). (d) Determine all fixed points of the linear transformation \(T: R^{2} \rightarrow R^{2}\) represented by T(x, y) = (y, x). Text Transcription: T: V rightarrow V T: R^2 rightarrow R^2

A translation in R2 is a function of the form T(x, y) = (x h, y k), where at least one of the constants h and k is nonzero. (a) Show that a translation in R2 is not a linear transformation. (b) For the translation T(x, y) = (x 2, y + 1), determine the images of (0, 0), (2, 1), and (5, 4). (c) Show that a translation in R2 has no fixed points.

Prove that (a) the zero transformation and (b) the identity transformation are linear transformations.

Let \(S = {v_{1}, v_{2}, v_{3}}\) be a set of linearly independent vectors in \(R^{3}\). Find a linear transformation T from \(R^{3}\) into \(R^{3}\) such that the set \({T(v_{1}), T(v_{2}), T(v_{3})}\) is linearly dependent. Text Transcription: S = {v_1, v_2, v_3} R^3 {T(v_1), T(v_2), T(v_3)}

Let \(S = {v_{1}, v_{2}, . . . , v_{n}}\) be a set of linearly dependent vectors in V, and let T be a linear transformation from V into V. Prove that the set \({T(v_{1}), T(v_{2}), . . . , T(v_{n})}\) is linearly dependent. Text Transcription: S = {v_1, v_2, . . . , v_n} {T(v_1), T(v_2), . . . , T(v_n)}

Let V be an inner product space. For a fixed vector \(v_{0}\) in V, define \(T: V \rightarrow R\) by \(T(v) = <v, v_{0}>\) . Prove that T is a linear transformation. Text Transcription: v_0 T: V rightarrow R T(v) = <v, v_0>

Define \(T: M_{n,n} \rightarrow R\) by \(T(A) = a_{11} + a_{22} + . . . + a_{nn}\) (the trace of A). Prove that T is a linear transformation. Text Transcription: T: M_n,n rightarrow R T(A) = a_11 + a_22 + cdot + a_nn

Let V be an inner product space with a subspace W having \(B = {w_{1}, w_{2}, . . . , w_{n}}\) as an orthonormal basis. Show that the function \(T: V \rightarrow W\) represented by \(T(v) = <v, w_{1}> w_{1} + <v, w_{2}> w_{2} + . . . + <v, w_{n}> w_{n} is a linear transformation. T is called the orthogonal projection of V onto W. Text Transcription: B = {w_1, w_2, . . . , w_n} T: V rightarrow W T(v) = <v, w_1> w_1 + <v, w_2> w_2 + cdot <v, w_n> w_n

Let \({v_{1}, v_{2}, . . . , v_{n}}\) be a basis for a vector space V. Prove that if a linear transformation \(T: V \rightarrow V\) satisfies \(T(v_{i}) = 0\) for i = 1, 2, . . . , n, then T is the zero transformation. Getting Started: To prove that T is the zero transformation, you need to show that T(v) = 0 for every vector v in V. (i) Let v be an arbitrary vector in V such that \(v = c_{1}v_{1} + c_{2}v_{2} + . . . + c_{n}v_{n}\). (ii) Use the definition and properties of linear transformations to rewrite T(v) as a linear combination of \(T(v_{i})\). (iii) Use the fact that \(T(v_{i}) = 0\) to conclude that T(v) = 0, making T the zero transformation. Text Transcription: {v_1, v_2, . . ., v_n} T: V rightarrow V T(v_i) = 0 v = c_1v_1 + c_2v_2 + cdot + c_n v_n T(v_i)

Guided Proof Prove that T: VW is a linear transformation if and only if T(au + bv) = aT(u) + bT(v) for all vectors u and v and all scalars a and b. Getting Started: This is an if and only if statement, so you need to prove the statement in both directions. To prove that T is a linear transformation, you need to show that the function satisfies the definition of a linear transformation. In the other direction, let T be a linear transformation. Use the definition and properties of a linear transformation to prove that T(au + bv) = aT(u) + bT(v). (i) Let T(au + bv) = aT(u) + bT(v). Show that T preserves the properties of vector addition and scalar multiplication by choosing appropriate values of a and b. (ii) To prove the statement in the other direction, assume that T is a linear transformation. Use the properties and definition of a linear transformation to show that T(au + bv) = aT(u) + bT(v)..