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True or False? In Exercises 43 and 44, 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.
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Elementary Linear Algebra (MindTap Course List)
- True or False? In Exercises 49 and 50, 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 To show that a set is not a vector space, it is sufficient to show that just one axiom is not satisfied. b The set of all first-degree polynomials with the standard operations is a vector space. c The set of all pairs of real numbers of the form (0,y), with the standard operations on R2, is a vector space.arrow_forwardProof Let W is a subspace of the vector space V. Prove that the zero vector in V is also the zero vector in W.arrow_forwardTrue or False? In Exercises 83-86, 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 n-tuples is n-space and is denoted by Rn. b The additive identity of a vector space is not unique. c Once a theorem has been proved for an abstract vector space, you need not give separate proofs for n-tuples, matrices, and polynomials.arrow_forward
- Verifying Subspaces In Exercises 1-6, verify that W is a subspace of V. In each case assume that V has the standard operations. W={(x,y,4x5y):xandyarerealnumbers}V=R3arrow_forwardSubsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R2 whose second component is the square of the first.arrow_forwardSubsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R3 whose components are nonnegative.arrow_forward
- Testing for a Vector Space In Exercises 13-36, determine whether the set, together with the standard operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails. The set of all first-degree polynomial functions ax,a0, whose graphs pass through the origin.arrow_forwardVerifying Subspaces In Exercises 1-6, verify that W is a subspace of V. In each case assume that V has the standard operations. W is the set of all 22 matrices of the form [0ab0] V=M2,2arrow_forwardSubsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all matrices in Mn,n with determinants equal to 1.arrow_forward
- Subsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R3 whose components are Pythagorean triples.arrow_forwardVerifying Subspaces In Exercises 1-6, verify that W is a subspace of V. In each case assume that V has the standard operations. W is the set of all 32 matrices of the form [aba2b00c] V=M3,2arrow_forwardTesting for a Vector SpaceIn Exercises 13-36, determine whether the set, together with the standard operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails. The set of all 22 diagonal matricesarrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning