Problem #4: Consider the following statements.Problem #4:(i) A vector space is a subspace of itself.(ii) If V is a vector space then there exists an x EV such that x+v - v for all v EV.(iii) M₂ x 2 is a subspace of M3 x 3Determine which of the above statements are True (1) or False (2).So, for example, if you think that the answers, in the above order, are True,False,False, then you would enter'1,2,2' into the answer box below (without the quotes).

Answered: Image /qna-images/answer/b3572fd3-fac2-4867-8c52-b59f810a3ea3.jpg

Answered: Problem #4: Consider the following…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.2: Linear Independence, Basis, And Dimension

Problem 15EQ

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Show that the three points (x1,y1)(x2,y2) and (x3,y3) in the a plane are collinear if and only if the matrix [x1y11x2y21x3y31] has rank less than 3.
Prove that if A is similar to B and A is diagonalizable, then B is diagonalizable.
How do I verify that Problem #8 is true? I think that is what it is asking for. Also, this question and many other questions in this section are very much conceptual. Although, the math can be used to verify if each question is true. This specific problem is from a linear algebra textbook called Linear Algebra w/ Applications and the author is by Jeffrey Holt. Right now, I'm in Section 7.1, which consists of what makes a vector space a vector space. Ironically, the author mentioned something new with an R raised to a matrix, as well as a P raised to a number. Nonetheless, I am most certainly not sure of how to answer the problem, but here are some pictures.
Which of the following are vector spaces? Justify your answer. (a) The set of all polynomials of degree 3. (b) The set of all vectors x = (x1, x2, x3), satisfying 3x₁ + 5x2 − 9x3 = 2023. (c) The set of all vectors x = (x1, x2, x3), satisfying 2024x₁ + x2 = 0 has a unique solution. (d) The set of all 3 × 3 matrices such that Ax= (e) The set of all n × n (n = N) diagonal matrices. - x3 = 0. -
by Problem 2. Consider the set V = R² with addition and scalar multiplication defined (X1, X2) ℗ (Y1, Y2) = (X1 + X2, Y₁ + y2), a (x₁, x₂) = (ax₁, x₂). Is V a vector space with these operations? Justify your answer.
1. a. b. x1 + x2 x1 2x1 T X2 Find the solution set of the following system of linear equations : 5x3 2x3 X3 || 3 1 0 What is the dimension of this set of solutions? Of what vector space is it a linear subspace? (R"?)
Problem 2. Let V = {(a₁, a₂): a₁, a2 € R}. For (a₁, a2), (b₁,b₂) € V and c E R, define (a₁, a2) + (b₁,b₂) = (a₁ + 2b₁, a2 +3b2) and c(a₁, a2) = (ca₁, ca₂). Is V a vector space over R with these operations? Justify your answer.
each statement is true or false. You do not need to give a reason. Problem 1. Consider the vector space P2 of real polynomials of degree at most 2. The set W = {p(x) | p(0) = 1} is a subspace of P2.
Problem 3. Determine whether the following sets form subspaces of the vector space R². (i) The set S₁ of all vectors (x₁, x2) = R² such that x1x2 = 0. (ii) The set S₂ of all vectors (x₁, x2) E R² such that x₁ = 3x2.
(3) For each of the following sets, determine whether it is linearly independent or dependent. (0) € 9000 3 (c) {x³x, 2x² + 4,-2x³ +3x²+2x+6} in P₂ (R), where P3 (R) denotes the vector space of polynomials over the reals of degree at most 3. -1 2 (d) {(-12 1) (1 7¹). (1¹3) (3¹2)} in M2x2(R), where M2x2(R) 0 denotes the vector space of 2 × matrices over the reals. {0·0)} span(S), i.e., orthogonal to every vector in span(S). (b) (4) Let S in R4. 7 in R4. CR4. Find all vectors in R4 that are orthogonal to (5) Let A and B be matrices over the reals. For each of the following statements, determine whether it is true or false. If it is true, prove it. If it is false, give a counter example to disprove it. (a) If A + B is defined, then rank(A + B) = rank(A) + rank(B). (b) If AB is defined, then rank(AB) = rank(A) rank(B). (c) If A has size m × n, then rank(A) ≤ min{m, n}. (d) rank(A) = rank(At) (e) nullity(A) = nullity(A¹)
Let B = {b1, b2} and C = {c1, c2} be two bases for a vector space V. You are given that b1 = 10c1 + 3c2, b2 = 27c1 + 8c2 and c1 = -8b1 + 3b2, c2 = 27b1 – 10b2. a B (i) If P = CB , then the value of B is k (ii) If P BC then the value of n is m (ii) If U = 4b1 + 3b2 and [u]c = |' | 4b1 + 3b2 and (ис- then the value of r is S
i only got two more

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