Concept explainers
True-False Review
For Questions (a)-(g), decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem from the text. If false, provide an example, illustration, or brief explanation of why the statement is false.
If
Want to see the full answer?
Check out a sample textbook solutionChapter 5 Solutions
Differential Equations and Linear Algebra (4th Edition)
Additional Math Textbook Solutions
Elementary & Intermediate Algebra
College Algebra with Modeling & Visualization (5th Edition)
College Algebra (6th Edition)
Linear Algebra and Its Applications (5th Edition)
A Graphical Approach to College Algebra (6th Edition)
- True or False? In Exercises 55 and 56, 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 A set S of vectors in an inner product space V is orthonormal when every vector is a unit vector and each pair of vectors is orthogonal. b If a set of nonzero vectors S in an inner product space V is orthogonal, then S is linearly independent.arrow_forwardCAPSTONE (a) Explain how to determine whether a function defines an inner product. (b) Let u and v be vectors in an inner product space V, such that v0. Explain how to find the orthogonal projection of u onto v.arrow_forwardProof Complete the proof of the cancellation property of vector addition by justifying each step. Prove that if u, v, and w are vectors in a vector space V such that u+w=v+w, then u=v. u+w=v+wu+w+(w)=v+w+(w)a._u+(w+(w))=v+(w+(w))b._u+0=v+0c._ u=vd.arrow_forward
- True or False? In Exercises 57and 58, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from 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) Two vectors in Rn are equal if and only if their corresponding components are equal. (b) The vector v is the additive identity of v.arrow_forwardTrue or False? In Exercises 57and 58, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from 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 subtract two vectors in Rn, subtract their corresponding components. (b) The zero vector 0 in Rn is the additive inverse of a vector.arrow_forwardProof Prove Theorem 4.12. THEOREM 4.12 Basis Tests in an n-Dimensional Space Let V be a vector space of dimension n. 1. If S={v1,v2,,vn} is a linearly independent set of vectors in V, then S is a basis for V. 2. If S={v1,v2,,vn} spans V, then S is a basis for V.arrow_forward
- True or false?In Exercises 85 and 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 dot product is the only inner product that can be defined in Rn. b A nonzero vector in an inner product can have a norm of zero.arrow_forwardProofProve in full detail that M2,2, with the standard operations, is a vector space.arrow_forwardTrue or false? In Exercises 43and 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. (a) The orthogonal complement of Rn is empty set. (b) If each vector vRn can be uniquely written as a sum of a vector s1 from S1 and a vector s2 from S2, then Rn is direct sum of S1 and S2.arrow_forward
- True or False?In Exercises 73 and 74, determine whether the 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 length or norm of a vector is v=|v1+v2+v3++vn|. b The dot product of two vectors u and v is another vector represented by uv=(u1v1,u2v2,u3v3,,unvn).arrow_forwardGuided Proof Prove that if w is orthogonal to each vector in S={v1,v2,,vn}, then w is orthogonal to every linear combination of vector in S. Getting Started: To prove that w is orthogonal to every linear combination of vectors in S, you need to show that their inner product is 0. i Write v as a linear combination of vectors, with arbitrary scalars c1,,cn in S. ii Form the inner product of w and v. iii Use the properties of inner products to rewrite the inner product w,v as a linear combination of the inner products w,vi, i=1,,n. iv Use the fact that w is orthogonal to each vector in S to lead to the conclusion that w is orthogonal to v.arrow_forwardProof Let u and v be a nonzero vectors in an inner product space V. Prove that uprojvu is orthogonal to v.arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning