Question

(linear algebra) 1. (5 points) Indicate whether each of the following statements are (T) true or (F) false. (a) T or F. If the set {u, v, w} is linearly independent, then u, v, and w are not in R2. (b) T or F. If T(x) = Ax, R(y) = By, and the composition R ◦ T is defined, then R ◦ T(x) = BAx. (c) T or F. If the set {u, v, w, x} spans Rm, then m ≤ 4. (d) T or F. Any set of vectors containing 0 is linearly dependent, including the set {0}. (e) T or F. If T is a matrix transformation, then T(0) = 0. 2. (10 points) Complete the following statement by selecting all options that apply. The columns of an m × n matrix A span Rm if ... (A) A has m pivot columns (a pivot in every row). (B) A has n pivot columns (a pivot in every column). (C) the equation Ax = 0 has only the trivial solution. (D) the equation Ax = 0 has nontrivial solutions. (E) for each b in Rm, the equation Ax = b has a solution. (F) there exists a b in Rm such that the equation Ax = b is inconsistent. (G) the associated system of equations has at least one free variable. (H) the associated system of equations has no free variables. (I) each b in Rm is a linear combination of the columns of A. (J) there exists a b in Rm that is not a linear combination of the columns of A. 3. (6 points) Let u = [8, 19], v = [4, 7], and w = [3, 5]. (a) Write u as a linear combination of v and w (find scalars a and b so that u = av + bw). (b) Does the set {u, v, w} span R2? 4. (6 points) Let T : R2 → R3 be a matrix transformation such that T(x) = [x1 - 3x2, x1 - x2, 3x1 - 4x2]. (a) Find a matrix A such that T(x) = Ax for each x in R2. (b) Find an x in R2 such that T(x) = [3, 7, 14]. 5. (12 points) Let T : R2 → R2 be a matrix transformation satisfying T([1, 0]) = [2, 1] and T([4, 3]) = [8, 2]. (a) What is T([5, 3])? (b) What is T([0, 1])? (c) Find a 2 × 2 matrix A for which T(x) = Ax. (d) What is T ◦ T ◦ T ◦ T([1, 0])? 

          (linear algebra)
1. (5 points) Indicate whether each of the following statements
are (T) true or (F) false.
(a) T or F. If the set {u, v, w} is linearly independent, then u, v,
and w are not in R2.
(b) T or F. If T(x) = Ax, R(y) = By, and the composition R ◦ T is defined,
then R ◦ T(x) = BAx.
(c) T or F. If the set {u, v, w, x} spans Rm, then m ≤ 4.
(d) T or F. Any set of vectors containing 0 is linearly dependent,
including the set {0}.
(e) T or F. If T is a matrix transformation, then T(0) = 0.

2. (10 points) Complete the following statement by selecting all
options that apply.
The columns of an m × n matrix A span Rm if ...
(A) A has m pivot columns (a pivot in every row).
(B) A has n pivot columns (a pivot in every column).
(C) the equation Ax = 0 has only the trivial solution.
(D) the equation Ax = 0 has nontrivial solutions.
(E) for each b in Rm, the equation Ax = b has a solution.
(F) there exists a b in Rm such that the equation Ax = b is
inconsistent.
(G) the associated system of equations has at least one free
variable.
(H) the associated system of equations has no free variables.
(I) each b in Rm is a linear combination of the columns of A.
(J) there exists a b in Rm that is not a linear combination of the
columns of A.

3. (6 points) Let u = [8, 19], v = [4, 7], and w = [3, 5].
(a) Write u as a linear combination of v and w (find scalars a and b
so that u = av + bw).
(b) Does the set {u, v, w} span R2?

4. (6 points) Let T : R2 → R3 be a matrix transformation such
that
T(x) = [x1 - 3x2, x1 - x2, 3x1 - 4x2].
(a) Find a matrix A such that T(x) = Ax for each x in R2.
(b) Find an x in R2 such that T(x) = [3, 7, 14].

5. (12 points) Let T : R2 → R2 be a matrix transformation
satisfying
T([1, 0]) = [2, 1] and T([4, 3]) = [8, 2].
(a) What is T([5, 3])?
(b) What is T([0, 1])?
(c) Find a 2 × 2 matrix A for which T(x) = Ax.
(d) What is T ◦ T ◦ T ◦ T([1, 0])?
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Added by Emilia C.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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(linear algebra) 1. (5 points) Indicate whether each of the following statements are (T) true or (F) false. (a) T or F. If the set {u, v, w} is linearly independent, then u, v, and w are not in R2. (b) T or F. If T(x) = Ax, R(y) = By, and the composition R ◦ T is defined, then R ◦ T(x) = BAx. (c) T or F. If the set {u, v, w, x} spans Rm, then m ≤ 4. (d) T or F. Any set of vectors containing 0 is linearly dependent, including the set {0}. (e) T or F. If T is a matrix transformation, then T(0) = 0. 2. (10 points) Complete the following statement by selecting all options that apply. The columns of an m × n matrix A span Rm if ... (A) A has m pivot columns (a pivot in every row). (B) A has n pivot columns (a pivot in every column). (C) the equation Ax = 0 has only the trivial solution. (D) the equation Ax = 0 has nontrivial solutions. (E) for each b in Rm, the equation Ax = b has a solution. (F) there exists a b in Rm such that the equation Ax = b is inconsistent. (G) the associated system of equations has at least one free variable. (H) the associated system of equations has no free variables. (I) each b in Rm is a linear combination of the columns of A. (J) there exists a b in Rm that is not a linear combination of the columns of A. 3. (6 points) Let u = [8, 19], v = [4, 7], and w = [3, 5]. (a) Write u as a linear combination of v and w (find scalars a and b so that u = av + bw). (b) Does the set {u, v, w} span R2? 4. (6 points) Let T : R2 → R3 be a matrix transformation such that T(x) = [x1 - 3x2, x1 - x2, 3x1 - 4x2]. (a) Find a matrix A such that T(x) = Ax for each x in R2. (b) Find an x in R2 such that T(x) = [3, 7, 14]. 5. (12 points) Let T : R2 → R2 be a matrix transformation satisfying T([1, 0]) = [2, 1] and T([4, 3]) = [8, 2]. (a) What is T([5, 3])? (b) What is T([0, 1])? (c) Find a 2 × 2 matrix A for which T(x) = Ax. (d) What is T ◦ T ◦ T ◦ T([1, 0])?
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For this exercise assume that all matrices are n x n. Each part of this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. An implication is False if there is an instance in which "statement 2" is false but "statement 1" is true. Complete parts (a) through (e). Justify each answer. a. If there is an n x n matrix D such that AD = I, then there is also an n x n matrix C such that CA = I. Choose the correct answer below. A. True; by the Invertible Matrix Theorem, if there is an n x n matrix D such that AD = I, then it must be true that there is also an n x n matrix C such that CA = I. B. True; by the Invertible Matrix Theorem if AD = I then A, D, or both is/are the identity matrix. Therefore, CA = I. C. False; it does not follow from the Invertible Matrix Theorem that if AD = I, then CA = I. D. False, because matrix multiplication is not commutative. It is possible that only D (and not A) is invertible in the equation AD = I. This implies that CA = I is only true when C is invertible (for cases where A is not invertible), but it is not given that C is invertible. b. If the columns of A are linearly independent, then the columns of A span R^n. Choose the correct answer below. A. True; by the Invertible Matrix Theorem if the columns of A are linearly dependent, then the columns of A must span R^n. B. True; by the Invertible Matrix Theorem if the columns of A are linearly independent, then the columns of A must span R^n. C. False; by the Invertible Matrix Theorem if the columns of A are linearly independent, then the columns of A do not span R^n. D. False; by the Invertible Matrix Theorem if the columns of A are linearly dependent, then the columns of A must span R^n. c. If the equation Ax = b has at least one solution for each b in R^n, then the solution is unique for each b. A. True, but only for x ≠ 0; by the Invertible Matrix Theorem if Ax = b has at least one solution for each b in R^n, then the equation Ax = 0 does not only have the trivial solution. B. False; by the Invertible Matrix Theorem if Ax = b has at least one solution for each b in R^n, then the linear transformation x → Ax does not map R^n onto R^n. C. False; by the Invertible Matrix Theorem if Ax = b has at least one solution for each b in R^n, then matrix A is not invertible. If A is not invertible, then according to the invertible matrix theorem the solution is not unique for each b. D. True; by the Invertible Matrix Theorem if Ax = b has at least one solution for each b in R^n, then matrix A is invertible. If A is invertible, then according to the invertible matrix theorem the solution is unique for each b. d. If the linear transformation (x) → Ax maps R^n into R^n, then A has n pivot positions. A. True; the linear transformation (x) → Ax will always map R^n into R^n for any n x n matrix. Therefore, according to the Invertible Matrix Theorem A has n pivot positions. B. False; according to the Invertible Matrix Theorem, (x) → Ax maps R^n into R^n, then A has n + 2 pivot positions. C. True; according to the Invertible Matrix Theorem if (x) → Ax maps R^n into R^n then A is invertible, and if a matrix is invertible it has n pivot positions. D. False; the linear transformation (x) → Ax will always map R^n into R^n for any n x n matrix. According to the Invertible Matrix Theorem A has n pivot positions only if (x) → Ax maps R^n onto R^n. e. If there is a b in R^n such that the equation Ax = b is inconsistent, then the transformation x → Ax is not one-to-one. A. True; according to the Invertible Matrix Theorem if there is a b in R^n such that the equation Ax = b is inconsistent, then matrix A is invertible. B. True; according to the Invertible Matrix Theorem if there is a b in R^n such that the equation Ax = b is inconsistent, then equation Ax = b does not have at least one solution for each b in R^n and this makes A not invertible. C. False; according to the Invertible Matrix Theorem if there is a b in R^n such that the equation Ax = b is inconsistent, then equation Ax = b has at least one solution for each b in R^n and this makes A invertible. D. False; according to the Invertible Matrix Theorem if there is a b in R^n such that the equation Ax = b is inconsistent, then the linear transformation x → Ax maps R^n onto R^n.

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Problem 6. Determine whether each of the following statements is true or false. Provide a justification for each of your responses. (a) If the coefficient matrix of a linear system has a pivot in the rightmost column, then the system is inconsistent. (b) If a linear system has two equations and four unknowns, then it must be consistent. (c) If a linear system having four equations and three unknowns is consistent, then the solution is unique. (d) Suppose that a linear system has four equations and four unknowns and that the coefficient matrix has four pivots. Then the linear system is consistent and has a unique solution. (e) Suppose that a linear system has five equations and three unknowns and that the coefficient matrix has a pivot in every column. Then the linear system is consistent and has a unique solution. Problem 7. Determine whether each of the following statements is true or false. Provide a justification for each of your responses. (a) Given two vectors v and w, the vector 0 is a linear combination of v and w. (b) Suppose that v1, v2, ..., vn is a collection of m-dimensional vectors and that the matrix [ v1 v2 ... vn ] has a pivot position in every row. Then every m-dimensional vector b can be written as a linear combination of v1, v2, ..., vn. (c) Suppose that v1, v2, ..., vn is a collection of m-dimensional vectors and that the matrix [ v1 v2 ... vn ] has a pivot position in every row and every column. Then every m-dimensional vector b can be written as a linear combination of v1, v2, ..., vn in exactly one way. (d) It is possible to find three 4-dimensional vectors v1, v2 and v3 such that every 4-dimensional vector can be written as a linear combination of v1, v2 and v3. Problem 8. Suppose that A = [ 3 -27 ; -2 18 ]. (a) Describe the solution space to the equation Ax = 0. (b) Find a 2 x 3 matrix B with no zero entries such that AB = 0.

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Transcript

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00:01 We have been given u is equals to 8, 19, that means use a matrix with two elements in it.
00:07 We have been given v over here and this is a matrix of 4 ,7, and we have been given w over here and this is a matrix of 3, 5.
00:16 Right...
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