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Chapter 4 1. For each of the following subsets of R3, determine if it is a subspace of R³: - (a) {(1, 2, 3) E R³: 2₁ + 3x2 x3 = 0}. (b) {(1, 2, 3) ER³: x1 + 4x2 + 2x3 = 3} (c) {(1, 2, 3) ER³: 11223 = (d) {(1, 2, 3) ER³: ₁ = 3x3} 2. Consider the following matrix: 0} (a) Is in NuLA? 1 (b) Is in ColA? A = 1 2 3 4 -2 3 1 6 1 2 4 6 3. Define T T(p) : P2 → R² by T(p) = [P] For example, if p(t) 一間 = (a) Show that T is a linear transformation. = 3+5t +712, then (b) Find a polynomial p in P2 that spans the kernel of T and describe the range of T. 4. For each of the following sets, determine if they are bases for R³. If not, do they span R3 and are they linearly independent? (a) 2. Use as few calculations as possible to determine if the following matrix is invertible: 4 10 20-3 -3 0 2 3. Assume 7: R2 R2 is a linear transformation where T(x1, x2)=(3x1-5x2, 2x1-x2). Show that T is invertible and find a formula for T-¹. Chapter 3 1. Find the following determinants. 1 2 1 (a) 0 3 2 (b) 1 2 3 1 2 1 2 -2 0 3 1 1 001 3 1 2 1 2. Which of the above matrices is invertible? Why? 3. Explain why, for a general 3 × 3 matrix, interchanging the first and the second row multiplies the determinant by -1. Answer in terms of the cofactor expansion. 4. If a matrix A is invertible, is 34 invertible? What is (3A)-¹? 5. Determine the values for the parameter s so that the system has a unique solution and describe the solution. sx₁ + 28x2 -2 3x1 +68x2 = 3 6. Compute the adjugate of the following matrix and use it to find its inverse. Check your work by multiplying A· A-¹ 3 4 10 1

Chapter 4 1. For each of the following subsets of R3, determine if it is a subspace of R³: - (a) {(1, 2, 3) E R³: 2₁ + 3x2 x3 = 0}. (b) {(1, 2, 3) ER³: x1 + 4x2 + 2x3 = 3} (c) {(1, 2, 3) ER³: 11223 = (d) {(1, 2, 3) ER³: ₁ = 3x3} 2. Consider the following matrix: 0} (a) Is in NuLA? 1 (b) Is in ColA? A = 1 2 3 4 -2 3 1 6 1 2 4 6 3. Define T T(p) : P2 → R² by T(p) = [P] For example, if p(t) 一間 = (a) Show that T is a linear transformation. = 3+5t +712, then (b) Find a polynomial p in P2 that spans the kernel of T and describe the range of T. 4. For each of the following sets, determine if they are bases for R³. If not, do they span R3 and are they linearly independent? (a) 2. Use as few calculations as possible to determine if the following matrix is invertible: 4 10 20-3 -3 0 2 3. Assume 7: R2 R2 is a linear transformation where T(x1, x2)=(3x1-5x2, 2x1-x2). Show that T is invertible and find a formula for T-¹. Chapter 3 1. Find the following determinants. 1 2 1 (a) 0 3 2 (b) 1 2 3 1 2 1 2 -2 0 3 1 1 001 3 1 2 1 2. Which of the above matrices is invertible? Why? 3. Explain why, for a general 3 × 3 matrix, interchanging the first and the second row multiplies the determinant by -1. Answer in terms of the cofactor expansion. 4. If a matrix A is invertible, is 34 invertible? What is (3A)-¹? 5. Determine the values for the parameter s so that the system has a unique solution and describe the solution. sx₁ + 28x2 -2 3x1 +68x2 = 3 6. Compute the adjugate of the following matrix and use it to find its inverse. Check your work by multiplying A· A-¹ 3 4 10 1

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Chapter 4 1. For each of the following subsets of R3, determine if it is a subspace of R³: - (a) {(1, 2, 3) E R³: 2₁ + 3x2 x3 = 0}. (b) {(1, 2, 3) ER³: x1 + 4x2 + 2x3 = 3} (c) {(1, 2, 3) ER³: 11223 = (d) {(1, 2, 3) ER³: ₁ = 3x3} 2. Consider the following matrix: 0} (a) Is in NuLA? 1 (b) Is in ColA? A = 1 2 3 4 -2 3 1 6 1 2 4 6 3. Define T T(p) : P2 → R² by T(p) = [P] For example, if p(t) 一間 = (a) Show that T is a linear transformation. = 3+5t +712, then (b) Find a polynomial p in P2 that spans the kernel of T and describe the range of T. 4. For each of the following sets, determine if they are bases for R³. If not, do they span R3 and are they linearly independent? (a)
Transcribed Image Text:Chapter 4 1. For each of the following subsets of R3, determine if it is a subspace of R³: - (a) {(1, 2, 3) E R³: 2₁ + 3x2 x3 = 0}. (b) {(1, 2, 3) ER³: x1 + 4x2 + 2x3 = 3} (c) {(1, 2, 3) ER³: 11223 = (d) {(1, 2, 3) ER³: ₁ = 3x3} 2. Consider the following matrix: 0} (a) Is in NuLA? 1 (b) Is in ColA? A = 1 2 3 4 -2 3 1 6 1 2 4 6 3. Define T T(p) : P2 → R² by T(p) = [P] For example, if p(t) 一間 = (a) Show that T is a linear transformation. = 3+5t +712, then (b) Find a polynomial p in P2 that spans the kernel of T and describe the range of T. 4. For each of the following sets, determine if they are bases for R³. If not, do they span R3 and are they linearly independent? (a)
2. Use as few calculations as possible to determine if the following matrix is invertible: 4 10 20-3 -3 0 2 3. Assume 7: R2 R2 is a linear transformation where T(x1, x2)=(3x1-5x2, 2x1-x2). Show that T is invertible and find a formula for T-¹. Chapter 3 1. Find the following determinants. 1 2 1 (a) 0 3 2 (b) 1 2 3 1 2 1 2 -2 0 3 1 1 001 3 1 2 1 2. Which of the above matrices is invertible? Why? 3. Explain why, for a general 3 × 3 matrix, interchanging the first and the second row multiplies the determinant by -1. Answer in terms of the cofactor expansion. 4. If a matrix A is invertible, is 34 invertible? What is (3A)-¹? 5. Determine the values for the parameter s so that the system has a unique solution and describe the solution. sx₁ + 28x2 -2 3x1 +68x2 = 3 6. Compute the adjugate of the following matrix and use it to find its inverse. Check your work by multiplying A· A-¹ 3 4 10 1
Transcribed Image Text:2. Use as few calculations as possible to determine if the following matrix is invertible: 4 10 20-3 -3 0 2 3. Assume 7: R2 R2 is a linear transformation where T(x1, x2)=(3x1-5x2, 2x1-x2). Show that T is invertible and find a formula for T-¹. Chapter 3 1. Find the following determinants. 1 2 1 (a) 0 3 2 (b) 1 2 3 1 2 1 2 -2 0 3 1 1 001 3 1 2 1 2. Which of the above matrices is invertible? Why? 3. Explain why, for a general 3 × 3 matrix, interchanging the first and the second row multiplies the determinant by -1. Answer in terms of the cofactor expansion. 4. If a matrix A is invertible, is 34 invertible? What is (3A)-¹? 5. Determine the values for the parameter s so that the system has a unique solution and describe the solution. sx₁ + 28x2 -2 3x1 +68x2 = 3 6. Compute the adjugate of the following matrix and use it to find its inverse. Check your work by multiplying A· A-¹ 3 4 10 1
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