Linear Algebra Problems Involving Fields, Vector Spaces, and Linear Maps
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This document provides an overview of exam questions and solutions related to linear algebra, specifically focusing on fields, vector spaces, and linear maps. It includes various problems that test comprehension of mathematical principles, such as analyzing complex number sets and determining properties of vector spaces and linear transformations. The exercise aims to assess the ability to apply theoretical knowledge and problem-solving skills in linear algebra and abstract algebra.
Linear Algebra Problems Involving Fields, Vector Spaces, and Linear Maps
- 1. LINEAR ALGEBRA PROBLEMS INVOLVING FIELDS, VECTOR SPACES, AND LINEAR MAPS EXAM SOLUTIONS For Any Exam Related Queries Reach us at: Email: - support@liveexamhelper.com Website: - https://www.liveexamhelper.com/
- 2. LINEAR ALGEBRA PROBLEMS INVOLVING FIELDS, VECTOR SPACES, AND LINEAR MAPS: EXAM SOLUTION Welcome to Linear algebra exam questions with answers from liveexamhelper.com. This set of exam questions explores various fundamental concepts in abstract algebra and linear algebra. The problems delve into topics such as the properties of fields, the structure of vector spaces, and the behavior of linear maps. You will be required to analyze complex sets, determine field properties, and investigate the dimensions of vector spaces and their corresponding linear transformations. These questions are designed to test your understanding of key mathematical principles and your ability to apply them to solve theoretical problems.
- 3. Question: Consider the set of complex numbers G = {a + bi | a, b Q}. ∈ (The G stands for Gauss; these numbers might be called Gaussian rational numbers, although I don’t know if they actually are.) Is G a field (with the same addition and multiplication operations as in C)? For a question like this, you should either explain why all the axioms for a field are satisfied (you can assume that they hold for C), or else explain why one of the axioms fails. Answer:
- 6. Question: Consider the set of complex numbers M = {re 2πiθ | r, θ Q}. ∈ Is M a field? The vector space V = (F2)2 has exactly four vectors (0, 0), (0, 1), (1, 0), and (1, 1); so V has exactly 24 = 16 subsets. How many of these 16 subsets are linearly independent? How many bases does V have? For a question like this, you might write some words explaining why some kinds of subset cannot possibly be linearly independent (“the vector (1, 1) is in the pay of Big Oil, and so cannot be part of any linearly independent set”). After this you might be left with just a few cases; you could perhaps say a few words about why each of these is or is not linearly independent.
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- 10. Question: Let V be the vector space of polynomials of degree at most five with real coefficients. Define a linear map T: V → R^3, T(p) = (p(1), p(2), p(3)). That is, the coordinates of the vector T(p) are the values of p at 1, 2, and 3. a) Find a basis of the null space of T. b) b) Find a basis of the range of T. Answer:
- 14. Question: Let V be the vector space of polynomials of degree at most 999 with real coefficients. Define a linear map T: V → R^100, T(p) = (p(1), p(2), . . . , p(100)). a) Find the dimension of the null space of T. b) Find the dimension of the range of T.
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- 17. Question: Let V be the vector space of polynomials of degree at most 99 with real coefficients. Define a linear map T: V → R^1000, T(p) = (p(1), p(2), . . . , p(1000)). a) Find the dimension of the null space of T. b) b) Find the dimension of the range of T. c) c) (This one is hard.) Is the vector (0, 1, 0, 1, 0, 1, . . . , 0, 1) in the range of T? That is, is there a polynomial of degree at most 99 whose values at 1, 2, . . . , 1000 alternate between 0 and 1? Solution:
- 20. Question: Give an example of a 3×3 matrix A of real numbers whose reduced row- echelon form is Solution:
- 23. CONCLUSION In conclusion, these exam questions aim to assess your comprehension of core concepts in fields, vector spaces, and linear maps. By examining the properties of complex number sets, the nature of vector spaces, and the effects of linear transformations, you are encouraged to apply both theoretical knowledge and problem-solving skills. Your responses should reflect a deep understanding of mathematical structures and the ability to analyze and derive meaningful results from abstract definitions and operations. This exercise is intended to demonstrate your proficiency in these areas and your capacity to think critically and analytically within the realm of linear algebra and abstract algebra.