Proof Let T: VW be a linear transformation. Provethat T is

Finding the Kernel of a Linear Transformation In Exercises 110, find the kernel of the linear transformation. . T: R3R3, T(x, y, z) = (0, 0, 0)

Finding the Kernel of a Linear Transformation In Exercises 110, find the kernel of the linear transformation. T: R3R3, T(x, y, z) = (x, 0, z)

Finding the Kernel of a Linear Transformation In Exercises 110, find the kernel of the linear transformation. T: R4R4, T(x, y, z, w) = (y, x, w, z

Finding the Kernel of a Linear Transformation In Exercises 110, find the kernel of the linear transformation. T: R3R3, T(x, y, z) = (z, y, x)

Finding the Kernel of a Linear Transformation In Exercises 110, find the kernel of the linear transformation. T: P3R, T(a0 + a1x + a2x2 + a3x3) = a1 + a2

Finding the Kernel of a Linear Transformation In Exercises 110, find the kernel of the linear transformation. T: P2R, T(a0 + a1x + a2x2) = a0

Finding the Kernel of a Linear Transformation In Exercises 110, find the kernel of the linear transformation. T: P2P1, T(a0 + a1x + a2x2) = a1 + 2a2x

Finding the Kernel of a Linear Transformation In Exercises 110, find the kernel of the linear transformation. T: P3P2, T(a0 + a1x + a2x2 + a3x3) = a1 + 2a2x + 3a3x2

Finding the Kernel of a Linear Transformation In Exercises 110, find the kernel of the linear transformation. T: R2R2, T(x, y) = (x + 2y, y x)

Finding the Kernel of a Linear Transformation In Exercises 110, find the kernel of the linear transformation. T: R2R2, T(x, y) = (x y, y x)

Finding the Kernel and Range In Exercises 1118, define the linear transformation T by T(x) = Ax. Find (a) the kernel of T and (b) the range of T. A = [ 1 3 2 4]

Finding the Kernel and Range In Exercises 1118, define the linear transformation T by T(x) = Ax. Find (a) the kernel of T and (b) the range of T. A = [ 1 3 2 6]

Finding the Kernel and Range In Exercises 1118, define the linear transformation T by T(x) = Ax. Find (a) the kernel of T and (b) the range of T. A = [ 1 0 1 1 2 2]

Finding the Kernel and Range In Exercises 1118, define the linear transformation T by T(x) = Ax. Find (a) the kernel of T and (b) the range of T. A = [ 1 0 2 2 1 1]

Finding the Kernel and Range In Exercises 1118, define the linear transformation T by T(x) = Ax. Find (a) the kernel of T and (b) the range of T. A = [ 1 1 2 3 3 2 ]

Finding the Kernel and Range In Exercises 1118, define the linear transformation T by T(x) = Ax. Find (a) the kernel of T and (b) the range of T. A = [ 1 1 0 1 2 1 ]

Finding the Kernel and Range In Exercises 1118, define the linear transformation T by T(x) = Ax. Find (a) the kernel of T and (b) the range of T. A = [ 1 3 4 1 2 1 3 2 1 2 1 1 4 1 3 1 ]

Finding the Kernel and Range In Exercises 1118, define the linear transformation T by T(x) = Ax. Find (a) the kernel of T and (b) the range of T. A = [ 1 2 2 3 3 1 2 5 2 1 0 1 4 0 0 ]

Finding the Kernel, Nullity, Range, and Rank In Exercises 1932, define the linear transformation T by T(x) = Ax. Find (a) ker(T), (b) nullity(T), (c) range(T), and (d) rank(T). A = [ 1 1 1 1]

Finding the Kernel, Nullity, Range, and Rank In Exercises 1932, define the linear transformation T by T(x) = Ax. Find (a) ker(T), (b) nullity(T), (c) range(T), and (d) rank(T). A = [ 3 9 2 6]

Finding the Kernel, Nullity, Range, and Rank In Exercises 1932, define the linear transformation T by T(x) = Ax. Find (a) ker(T), (b) nullity(T), (c) range(T), and (d) rank(T). A = [ 5 1 1 3 1 1 ]

Finding the Kernel, Nullity, Range, and Rank In Exercises 1932, define the linear transformation T by T(x) = Ax. Find (a) ker(T), (b) nullity(T), (c) range(T), and (d) rank(T). A = [ 4 0 2 1 0 3 ]

Finding the Kernel, Nullity, Range, and Rank In Exercises 1932, define the linear transformation T by T(x) = Ax. Find (a) ker(T), (b) nullity(T), (c) range(T), and (d) rank(T). A = [ 9 10 3 10 3 10 1 10]

Finding the Kernel, Nullity, Range, and Rank In Exercises 1932, define the linear transformation T by T(x) = Ax. Find (a) ker(T), (b) nullity(T), (c) range(T), and (d) rank(T). A = [ 1 26 5 26 5 26 25 26]

Finding the Kernel, Nullity, Range, and Rank In Exercises 1932, define the linear transformation T by T(x) = Ax. Find (a) ker(T), (b) nullity(T), (c) range(T), and (d) rank(T). A = [ 1 0 1 0 1 0 1 0 1 ]

Finding the Kernel, Nullity, Range, and Rank In Exercises 1932, define the linear transformation T by T(x) = Ax. Find (a) ker(T), (b) nullity(T), (c) range(T), and (d) rank(T). A = [ 1 0 0 0 0 0 0 0 1 ]

Finding the Kernel, Nullity, Range, and Rank In Exercises 1932, define the linear transformation T by T(x) = Ax. Find (a) ker(T), (b) nullity(T), (c) range(T), and (d) rank(T). A = [ 4 9 4 9 2 9 4 9 4 9 2 9 2 9 2 9 1 9 ]

Finding the Kernel, Nullity, Range, and Rank In Exercises 1932, define the linear transformation T by T(x) = Ax. Find (a) ker(T), (b) nullity(T), (c) range(T), and (d) rank(T). . A = [ 1 3 2 3 1 3 2 3 1 3 2 3 1 3 2 3 1 3 ]

Finding the Kernel, Nullity, Range, and Rank In Exercises 1932, define the linear transformation T by T(x) = Ax. Find (a) ker(T), (b) nullity(T), (c) range(T), and (d) rank(T). A = [ 0 4 2 0 3 11]

Finding the Kernel, Nullity, Range, and Rank In Exercises 1932, define the linear transformation T by T(x) = Ax. Find (a) ker(T), (b) nullity(T), (c) range(T), and (d) rank(T). A = [ 1 0 1 0 0 1 0 1

Finding the Kernel, Nullity, Range, and Rank In Exercises 1932, define the linear transformation T by T(x) = Ax. Find (a) ker(T), (b) nullity(T), (c) range(T), and (d) rank(T). A = [ 2 1 3 6 2 1 3 6 3 1 5 2 1 1 0 4 13 1 14 16]

Finding the Kernel, Nullity, Range, and Rank In Exercises 1932, define the linear transformation T by T(x) = Ax. Find (a) ker(T), (b) nullity(T), (c) range(T), and (d) rank(T). A = [ 3 4 2 2 3 3 6 8 4 1 10 4 15 14 20]

Finding the Nullity and Describing the Kernel and Range In Exercises 3340, let T: R3R3 be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. rank(T) = 2

Finding the Nullity and Describing the Kernel and Range In Exercises 3340, let T: R3R3 be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. rank(T) = 1

Finding the Nullity and Describing the Kernel and Range In Exercises 3340, let T: R3R3 be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. rank(T) = 0

Finding the Nullity and Describing the Kernel and Range In Exercises 3340, let T: R3R3 be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. rank(T) = 3

Finding the Nullity and Describing the Kernel and Range In Exercises 3340, let T: R3R3 be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. T is the counterclockwise rotation of 45 about the z-axis: T(x, y, z) = ( 2 2 x 2 2 y, 2 2 x + 2 2 y, z)

Finding the Nullity and Describing the Kernel and Range In Exercises 3340, let T: R3R3 be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. T is the reflection through the yz-coordinate plane: T(x, y, z) = (x, y, z)

Finding the Nullity and Describing the Kernel and Range In Exercises 3340, let T: R3R3 be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. T is the projection onto the vector v = (1, 2, 2): T(x, y, z) = x + 2y + 2z 9 (1, 2, 2)

Finding the Nullity and Describing the Kernel and Range In Exercises 3340, let T: R3R3 be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. T is the projection onto the xy-coordinate plane: T(x, y, z) = (x, y, 0)

Finding the Nullity of a Linear Transformation In Exercises 4146, find the nullity of T. T: R4R2, rank(T) = 2

Finding the Nullity of a Linear Transformation In Exercises 4146, find the nullity of T. T: R4R4, rank(T) = 0

Finding the Nullity of a Linear Transformation In Exercises 4146, find the nullity of T. T: P5P2, rank(T) = 3

Finding the Nullity of a Linear Transformation In Exercises 4146, find the nullity of T. T: P3P1, rank(T) = 2

Finding the Nullity of a Linear Transformation In Exercises 4146, find the nullity of T. T: M2,4M4,2, rank(T) = 4

Finding the Nullity of a Linear Transformation In Exercises 4146, find the nullity of T. T: M3,3M2,3, rank(T) = 6

Verifying That T Is One-to-One and Onto In Exercises 4750, verify that the matrix defines a linear function T that is one-to-one and onto. A = [ 2 0 0 2]

Verifying That T Is One-to-One and Onto In Exercises 4750, verify that the matrix defines a linear function T that is one-to-one and onto. A = [ 1 0 0 1]

Verifying That T Is One-to-One and Onto In Exercises 4750, verify that the matrix defines a linear function T that is one-to-one and onto. A = [ 1 0 0 0 0 1 0 1 0 ]

Verifying That T Is One-to-One and Onto In Exercises 4750, verify that the matrix defines a linear function T that is one-to-one and onto. A = [ 1 1 0 2 2 4 3 4 1 ]

Determining Whether T Is One-to-One, Onto, or Neither In Exercises 5154, determine whether the linear transformation is one-to-one, onto, or neither. T in Exercise 3

Determining Whether T Is One-to-One, Onto, or Neither In Exercises 5154, determine whether the linear transformation is one-to-one, onto, or neither. T in Exercise 10

Determining Whether T Is One-to-One, Onto, or Neither In Exercises 5154, determine whether the linear transformation is one-to-one, onto, or neither. T: R2R3, T(x) = Ax, where A is given in Exercise 21

Determining Whether T Is One-to-One, Onto, or Neither In Exercises 5154, determine whether the linear transformation is one-to-one, onto, or neither. T: R5R3, T(x) = Ax, where A is given in Exercise 18

Identify the zero element and standard basis for each of the isomorphic vector spaces in Example 12.

Which vector spaces are isomorphic to R6? (a) M2,3 (b) P6 (c) C[0, 6] (d) M6,1 (e) P5 (f) C[3, 3] (g) {(x1, x2, x3, 0, x5, x6, x7): xi is a real number}

Calculus Define T: P4P3 by T(p) = p. What is the kernel of T?

Calculus Define T: P2R by T(p) = 1 0 p(x) dx. What is the kernel of T?

Let T: R3R3 be the linear transformation that projects u onto v = (2, 1, 1). (a) Find the rank and nullity of T. (b) Find a basis for the kernel of T.

CAPSTONE Let T: R4R3 be the linear transformation represented by T(x) = Ax, where A = [ 1 0 0 2 1 0 1 2 0 0 3 1 ] . (a) Find the dimension of the domain. (b) Find the dimension of the range. (c) Find the dimension of the kernel. (d) Is T one-to-one? Explain. (e) Is T onto? Explain. (f) Is T an isomorphism? Explain.

For the transformation T: RnRn represented by T(x) = Ax, what can be said about the rank of T when (a) det(A) 0 and (b) det(A) = 0?

Writing Let T: RmRn be a linear transformation. Explain the differences between the concepts of one-to-one and onto. What can you say about m and n when T is onto? What can you say about m and n when T is one-to-one?

Define T: Mn,nMn,n by T(A) = A AT. Show that the kernel of T is the set of n n symmetric matrices.

Determine a relationship among m, n, j, and k such that Mm,n is isomorphic to Mj,k.

True or False? In Exercises 65 and 66, 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 set of all vectors mapped from a vector space V into another vector space W by a linear transformation T is the kernel of T. (b) The range of a linear transformation from a vector space V into a vector space W is a subspace of V. (c) The vector spaces R3 and M3,1 are isomorphic to each other.

True or False? In Exercises 65 and 66, 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 dimension of a linear transformation T from a vector space V into a vector space W is the rank of T. (b) A linear transformation T from V into W is one-to-one when the preimage of every w in the range consists of a single vector v. (c) The vector spaces R2 and P1 are isomorphic to each other.

Guided Proof Let B be an invertible n n matrix. Prove that the linear transformation T: Mn,nMn,n represented by T(A) = AB is an isomorphism. Getting Started: To show that the linear transformation is an isomorphism, you need to show that T is both onto and one-to-one. (i) T is a linear transformation with vector spaces of equal dimension, so by Theorem 6.8, you only need to show that T is one-to-one. (ii) To show that T is one-to-one, you need to determine the kernel of T and show that it is {0} (Theorem 6.6). Use the fact that B is an invertible n n matrix and that T(A) = AB. (iii) Conclude that T is an isomorphism.

Proof Let T: VW be a linear transformation. Prove that T is one-to-one if and only if the rank of T equals the dimension of V

Proof Prove Theorem 6.7.

Proof Let T: VW be a linear transformation, and let U be a subspace of W. Prove that the set T 1(U) = {v V: T(v) U} is a subspace of V. What is T 1(U) when U = {0}?

In Exercises 1–10, find the kernel of the linear transformation. \(T: R^{3} \rightarrow R^{3}, T(x, y, z) = (0, 0, 0)\) Text Transcription: T: R^3 rightarrow R^3, T(x, y, z) = (0, 0, 0)

In Exercises 1–10, find the kernel of the linear transformation. \(T: R^{3} \rightarrow R^{3}, T(x, y, z) = (x, 0, z)\) Text Transcription: T: R^3 rightarrow R^3, T(x, y, z) = (x, 0, z)

In Exercises 1–10, find the kernel of the linear transformation. \(T: R^{4} \rightarrow R^{4}, T(x, y, z, w) = (y, x, w, z)\) Text Transcription: T: R^4 rightarrow R^4, T(x, y, z, w) = (y, x, w, z)

In Exercises 1–10, find the kernel of the linear transformation. \(T: R^{3} \rightarrow R^{3}, T(x, y, z) = (?z, ?y, ?x)\) Text Transcription: T: R^3 rightarrow R^3, T(x, y, z) = (?z, ?y, ?x)

In Exercises 1–10, find the kernel of the linear transformation. \(T: P_{3} \rightarrow R T(a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3}) = a_{1} + a_{2}\) Text Transcription: T: P_3 rightarrow R, T(a_0 + a_1x + a_2x^2 a_3x^3) = a_1 +a_2

In Exercises 1–10, find the kernel of the linear transformation. \(T: P_{2} \rightarrow R, T(a_{0} + a_{1}x + a_{2}x^{2}) = a_{0}\) Text Transcription: T: P_2 rightarrow R, T(a_0 + a_1x + a_2x^2) = a_0

In Exercises 1–10, find the kernel of the linear transformation. \(T: P_{2} \rightarrow P_{1}, T(a_{0} + a_{1}x + a_{2}x^{2}) = a_{1} + 2a_{2}x\) Text Transcription: T: P_2 rightarrow P_1, T(a_0 + a_1x + a_2x^2) = a_1 + 2a_2x

In Exercises 1–10, find the kernel of the linear transformation. \(T: P_3 \rightarrow P_2, T(a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3}) = a_{1} + 2a_{2}x + 3a_{3}x^{2}\) Text Transcription: T: P_3 rightarrow P_2, T(a_0 + a_1x + a_2x^2 + a_3x^3) = a_1 + 2a_2x + 3a_3x^2

In Exercises 1–10, find the kernel of the linear transformation. \(T: R^{2} \rightarrow R^{2}, T(x, y) = (x + 2y, y ? x)\) Text Transcription: T: R^2 rightarrow R^2, T(x, y) = (x + 2y, y ? x)

In Exercises 1–10, find the kernel of the linear transformation. \(T: R^{2} \rightarrow R^{2}, T(x, y) = (x ? y, y ? x)\) Text Transcription: T: R^2 rightarrow R^2, T(x, y) = (x ? y, y ? x)

In Exercises 11–18, define the linear transformation T by T(x_ = Ax. Find (a) the kernel of T and (b) the range of T. \(A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right] \) Text Transcription: A = [_3^1 _4^2]

In Exercises 11–18, define the linear transformation T by T(x_ = Ax. Find (a) the kernel of T and (b) the range of T. \(A=\left[\begin{array}{rr} 1 & 2 \\ -3 & -6 \end{array}\right] \) Text Transcription: A = [_-3^1 _-6^2]

In Exercises 11–18, define the linear transformation T by T(x_ = Ax. Find (a) the kernel of T and (b) the range of T. \(A=\left[\begin{array}{rrr} 1 & -1 & 2 \\ 0 & 1 & 2 \end{array}\right] \) Text Transcription: A = [_0^1 _1^-1 _2^2]

In Exercises 11–18, define the linear transformation T by T(x_ = Ax. Find (a) the kernel of T and (b) the range of T. \(A=\left[\begin{array}{rrr} 1 & -2 & 1 \\ 0 & 2 & 1 \end{array}\right] \) Text Transcription: A = [_0^1 _2^-2 _1^1]

In Exercises 11–18, define the linear transformation T by T(x_ = Ax. Find (a) the kernel of T and (b) the range of T. \(A=\left[\begin{array}{rr} 1 & 3 \\ -1 & -3 \\ 2 & 2 \end{array}\right] \) Text Transcription: A = [_2^-1^1 _2^-3^3]

In Exercises 11–18, define the linear transformation T by T(x_ = Ax. Find (a) the kernel of T and (b) the range of T. \(A=\left[\begin{array}{rr} 1 & 1 \\ -1 & 2 \\ 0 & 1 \end{array}\right] \) Text Transcription: A = [_0^-1^1 _1^2^1]

In Exercises 11–18, define the linear transformation T by T(x_ = Ax. Find (a) the kernel of T and (b) the range of T. \(A=\left[\begin{array}{rrrr} 1 & 2 & -1 & 4 \\ 3 & 1 & 2 & -1 \\ -4 & -3 & -1 & -3 \\ -1 & -2 & 1 & 1 \end{array}\right] \) Text Transcription: A = [_-1^-4^3^1 _-2^-3^1^2 _1^-1^2^-1 _1^-3^-1^4]

In Exercises 11–18, define the linear transformation T by T(x_ = Ax. Find (a) the kernel of T and (b) the range of T. \(A=\left[\begin{array}{rrrrr} -1 & 3 & 2 & 1 & 4 \\ 2 & 3 & 5 & 0 & 0 \\ 2 & 1 & 2 & 1 & 0 \end{array}\right] \) Text Transcription: A = [_2^2^-1 _1^3^3 _2^5^2 _1^0^1 _0^0^4]

In Exercises 19–32, define the linear transformation T by T(x) = Ax. Find (a) kerT, (b) nullityT, (c) rangeT, and (d) rankT. \(A=\left[\begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array}\right] \) Text Transcription: A = [_1^-1 _1^1]

In Exercises 19–32, define the linear transformation T by T(x) = Ax. Find (a) kerT, (b) nullityT, (c) rangeT, and (d) rankT. \(A=\left[\begin{array}{rr} 3 & 2 \\ -9 & -6 \end{array}\right] \) Text Transcription: A = [_-9^3 _-6^2]

In Exercises 19–32, define the linear transformation T by T(x) = Ax. Find (a) kerT, (b) nullityT, (c) rangeT, and (d) rankT. \(A=\left[\begin{array}{rr} 5 & -3 \\ 1 & 1 \\ 1 & -1 \end{array}\right] \) Text Transcription: A = [_1^1^5 _-1^1^-3]

In Exercises 19–32, define the linear transformation T by T(x) = Ax. Find (a) kerT, (b) nullityT, (c) rangeT, and (d) rankT. \(A=\left[\begin{array}{rr} 4 & 1 \\ 0 & 0 \\ 2 & -3 \end{array}\right] \) Text Transcription: A = [_2^0^4 _-3^0^1]

In Exercises 19–32, define the linear transformation T by T(x) = Ax. Find (a) kerT, (b) nullityT, (c) rangeT, and (d) rankT. \(A=\left[\begin{array}{cc} \frac{9}{10} & \frac{3}{10} \\ \frac{3}{10} & \frac{1}{10} \end{array}\right] \) Text Transcription: A = [_3/10^9/10 1/10^3/10]

In Exercises 19–32, define the linear transformation T by T(x) = Ax. Find (a) kerT, (b) nullityT, (c) rangeT, and (d) rankT. \(A=\left[\begin{array}{rr} \frac{1}{26} & -\frac{5}{26} \\ -\frac{5}{26} & \frac{25}{26} \end{array}\right] \) Text Transcription: A = [_-5/26^1/26 _25/26^-5/26]

In Exercises 19–32, define the linear transformation T by T(x) = Ax. Find (a) kerT, (b) nullityT, (c) rangeT, and (d) rankT. \(A=\left[\begin{array}{lll} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{array}\right] \) Text Transcription: A = [_1^0^1 _0^1^0 _1^0^1]

In Exercises 19–32, define the linear transformation T by T(x) = Ax. Find (a) kerT, (b) nullityT, (c) rangeT, and (d) rankT. \(A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right] \) Text Transcription: A = [_0^0^1 _0^0^0 _1^0^0]

In Exercises 19–32, define the linear transformation T by T(x) = Ax. Find (a) kerT, (b) nullityT, (c) rangeT, and (d) rankT. \(A=\left[\begin{array}{rrr} \frac{4}{9} & -\frac{4}{9} & \frac{2}{9} \\ -\frac{4}{9} & \frac{4}{9} & -\frac{2}{9} \\ \frac{2}{9} & -\frac{2}{9} & \frac{1}{9} \end{array}\right] \) Text Transcription: A = [_2/9^-4/9^4/9 _-2/9^4/9^-4/9 _1/9^-2/9^2/9]

In Exercises 19–32, define the linear transformation T by T(x) = Ax. Find (a) kerT, (b) nullityT, (c) rangeT, and (d) rankT. \(A=\left[\begin{array}{rrr} -\frac{1}{3} & \frac{2}{3} & -\frac{1}{3} \\ \frac{2}{3} & \frac{1}{3} & \frac{2}{3} \\ -\frac{1}{3} & \frac{2}{3} & -\frac{1}{3} \end{array}\right] \) Text Transcription: A = [_-1/3^2/3^-1/3 _2/3^1/3^2/3 _-1/3^2/3^-1/3]

In Exercises 19–32, define the linear transformation T by T(x) = Ax. Find (a) kerT, (b) nullityT, (c) rangeT, and (d) rankT. \(A=\left[\begin{array}{rrr} 0 & -2 & 3 \\ 4 & 0 & 11 \end{array}\right] \) Text Transcription: A = [_4^0 _0^-2 _11^3]

In Exercises 19–32, define the linear transformation T by T(x) = Ax. Find (a) kerT, (b) nullityT, (c) rangeT, and (d) rankT. \(A=\left[\begin{array}{llll} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{array}\right] \) Text Transcription: A = [_0^1 _0^1 _1^0 _1^0]

In Exercises 19–32, define the linear transformation T by T(x) = Ax. Find (a) kerT, (b) nullityT, (c) rangeT, and (d) rankT. \(A=\left[\begin{array}{rrrrr} 2 & 2 & -3 & 1 & 13 \\ 1 & 1 & 1 & 1 & -1 \\ 3 & 3 & -5 & 0 & 14 \\ 6 & 6 & -2 & 4 & 16 \end{array}\right] \) Text Transcription: A = [_6^3^1^2 _6^3^1^2 _-2^-5^1^-1 _4^0^1^1 _16^14^-1^13]

In Exercises 19–32, define the linear transformation T by T(x) = Ax. Find (a) kerT, (b) nullityT, (c) rangeT, and (d) rankT. \(A=\left[\begin{array}{rrrrr} 3 & -2 & 6 & -1 & 15 \\ 4 & 3 & 8 & 10 & -14 \\ 2 & -3 & 4 & -4 & 20 \end{array}\right] \) Text Transcription: A = [_2^4^3 _-3^3^-2 _4^8^6 _-4^10^-1 _20^-14^15]

In Exercises 33–40, let \(T: R^{3] \rightarrow R^{3}\) be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. rank(T) = 2 Text Transcription: T: R^3 rightarrow R^3

In Exercises 33–40, let \(T: R^{3] \rightarrow R^{3}\) be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. rank(T) = 1 Text Transcription: T: R^3 rightarrow R^3

In Exercises 33–40, let \(T: R^{3] \rightarrow R^{3}\) be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. rank(T) = 0 Text Transcription: T: R^3 rightarrow R^3

In Exercises 33–40, let \(T: R^{3] \rightarrow R^{3}\) be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. rank(T) = 3 Text Transcription: T: R^3 rightarrow R^3

Finding the Nullity and Describing the Kernel and Range In Exercises 3340, let T: R3R3 be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. T is the counterclockwise rotation of 45 about the z-axis: T(x, y, z) = ( 2 2 x 2 2 y, 2 2 x + 2 2 y, z)

T is the reflection through the yz-coordinate plane: T(x, y, z) = (?x, y, z)

Finding the Nullity and Describing the Kernel and Range In Exercises 3340, let T: R3R3 be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. T is the projection onto the vector v = (1, 2, 2): T(x, y, z) = x + 2y + 2z 9 (1, 2, 2)

T is the projection onto the xy-coordinate plane: T(x, y, z) = (x, y, 0)

Exercises 41–46, find the nullity of T. \(T: R^{4} \rightarrow R^{2}\), rank(T) = 2 Text Transcription: T: R^4 rightarrow R^2

Exercises 41–46, find the nullity of T. \(T: R^{4} \rightarrow R^{4}\), rank(T) = 0 Text Transcription: T: R^4 rightarrow R^4

Exercises 41–46, find the nullity of T. \(T: P_{5} \rightarrow P_{2}\), rank(T) = 3 Text Transcription: T: P_5 rightarrow P_2

Exercises 41–46, find the nullity of T. \(T: P_{3} \rightarrow P_{1}\), rank(T) = 2 Text Transcription: T: P_5 rightarrow P_2

Exercises 41–46, find the nullity of T. \(I: M_{2,4} \rightarrow M_{4,2}\), rank(T) = 4 Text Transcription: T: M_2,4 rightarrow M_4,2

Exercises 41–46, find the nullity of T. \(I: M_{3,3} \rightarrow M_{2,3}\), rank(T) = 6 Text Transcription: T: M_3,3 rightarrow M_2,3

In Exercises 47–50, verify that the matrix defines a linear function T that is one-to-one and onto. \(A=\left[\begin{array}{rr} -2 & 0 \\ 0 & 2 \end{array}\right] \) Text Transcription: A = [_0^-2 _2^0]

In Exercises 47–50, verify that the matrix defines a linear function T that is one-to-one and onto. \(A=\left[\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right] \) Text Transcription: A = [_0^1 _-1^0]

In Exercises 47–50, verify that the matrix defines a linear function T that is one-to-one and onto. \(A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right] \) Text Transcription: A = [_0^0^1 _1^0^0 _0^1^0]

Verifying That T Is One-to-One and Onto In Exercises 4750, verify that the matrix defines a linear function T that is one-to-one and onto. A = [ 1 1 0 2 2 4 3 4 1 ]

In Exercises 51–54, determine whether the linear transformation is one-to-one, onto, or neither. T in Exercise 3

In Exercises 51–54, determine whether the linear transformation is one-to-one, onto, or neither. T in Exercise 10

In Exercises 51–54, determine whether the linear transformation is one-to-one, onto, or neither. \(T: R^{2} \rightarrow R^{3}, T(x) = Ax\), where A is given in Exercise 21 Text Transcription: T: R^2 rightarrow R^3, T(x) = Ax

Determining Whether T Is One-to-One, Onto, or Neither In Exercises 5154, determine whether the linear transformation is one-to-one, onto, or neither. T: R5R3, T(x) = Ax, where A is given in Exercise 18

Identify the zero element and standard basis for each of the isomorphic vector spaces in Example 12.

Which vector spaces are isomorphic to \(R^{6}\)? (a) \(M_{2,3}\) (b) \(P_{6}\) (c) C[0, 6] (d) \(M_{6,1}\) (e) \(P_{5}\) (f) C’[?3, 3] (g) {\((x_{1}, x_{2}, x_{3}, 0, x_{5}, x_{6}, x_{7}): x_{i}\) is a real number} Text Transcription: R^6 M_2,3 P_6 M_6,1 P_5 (x_1, x_2, x_3, 0, x_5, x_6, x_7): x_i

Define \(T: P_{4} \rightarrow P_{3}\) by T(p) = p’. What is the kernel of T? Text Transcription: T: P_4 rightarrow P_3

Define \(T: P_{2}R\) by \(T(p)=\int_{0}^{1} p(x) d x\). What is the kernel of T? Text Transcription: T: P_2 rightarrow R T(p) = integ _0^1 p(x) dx

Let \(T: R^{3} \rightarrow R^{3}\( be the linear transformation that projects u onto v = (2, ?1, 1). (a) Find the rank and nullity of T. (b) Find a basis for the kernel of T. Text Transcription: T: R^3 rightarrow R^3

Let \(T: R^{4} \rightarrow R^{3}\) be the linear transformation represented by T(x) = Ax, where \(A=\left[\begin{array}{rrrr} 1 & -2 & 1 & 0 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 1 \end{array}\right] \) . (a) Find the dimension of the domain. (b) Find the dimension of the range. (c) Find the dimension of the kernel. (d) Is T one-to-one? Explain. (e) Is T onto? Explain. (f ) Is T an isomorphism? Explain. Text Transcription: T: R^4 rightarrow R^3 A = [_0^0^1 _0^1^-2 _0^2^1 _1^3^0]

For the transformation \(T: R^{n} \rightarrow R^{n}\) represented by T(x) = Ax, what can be said about the rank of T when (a) \(det(A) \neq 0\) and (b) det(A) = 0? Text Transcription: T: R^n rightarrow R^n det(A) neq 0

Let \(T: R^{m} \rightarrow R^{n}\) be a linear transformation. Explain the differences between the concepts of one-to-one and onto. What can you say about m and n when T is onto? What can you say about m and n when T is one-to-one? Text Transcription: T; R^m rightarrow R^n

Define \(T: M_{n,n} \rightarrow M_{n,n}\) by \(T(A) = A ? A^{T}\). Show that the kernel of T is the set of n × n symmetric matrices. Text Transcription: T: M_n,n rightarrow M_n,n T(A) = A - A^T

Determine a relationship among m, n, j, and k such that \(M_{m,n}\) is isomorphic to \(M_{j,k}\). Text Transcription: M_m,n M_j,k

In Exercises 65 and 66, 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 set of all vectors mapped from a vector space V into another vector space W by a linear transformation T is the kernel of T. (b) The range of a linear transformation from a vector space V into a vector space W is a subspace of V. (c) The vector spaces \(R^{3} and \(M_{3,1}\( are isomorphic to each other. Text Transcription: R^3 M_3,1

In Exercises 65 and 66, 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 dimension of a linear transformation T from a vector space V into a vector space W is the rank of T. (b) A linear transformation T from V into W is one-to-one when the preimage of every w in the range consists of a single vector v. (c) The vector spaces \(R_{2}\) and \(P_{1}\) are isomorphic to each other. Text Transcription: R^2 P_1

Let B be an invertible n × n matrix. Prove that the linear transformation \(T: M_{n,n} \rightarrow M_{n,n}\) represented by T(A) = AB is an isomorphism. Getting Started: To show that the linear transformation is an isomorphism, you need to show that T is both onto and one-to-one. (i) T is a linear transformation with vector spaces of equal dimension, so by Theorem 6.8, you only need to show that T is one-to-one. (ii) To show that T is one-to-one, you need to determine the kernel of T and show that it is {0} (Theorem 6.6). Use the fact that B is an invertible n × n matrix and that T(A) = AB. (iii) Conclude that T is an isomorphism. Text Transcription: T: M_n,n rightarrow M_n,n

Proof Let T: VW be a linear transformation. Prove that T is one-to-one if and only if the rank of T equals the dimension of V

Prove Theorem 6.7.

Proof Let T: VW be a linear transformation, and let U be a subspace of W. Prove that the set T 1(U) = {v V: T(v) U} is a subspace of V. What is T 1(U) when U = {0}?