Guided Proof Prove that if
Getting Started: To prove that
(i) | Write
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(ii) | Form the inner product of
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(iii) | Use the properties of inner products to rewrite the inner product
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(iv) | Use the fact that
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Elementary Linear Algebra (MindTap Course List)
- Proof When V is spanned by {v1,v2,...,vk} and one of these vector can be written as a linear combination of the other k1 vectors, prove that the span of these k1 vector is also V.arrow_forwardGuided Proof Prove that a nonempty subset of a finite set of linearly independent vectors is linearly independent. Getting Started: You need to show that a subset of a linearly independent set of vectors cannot be linearly dependent. (i) Assume S is a set of linearly independent vectors. Let T be a subset of S. (ii) If T is linearly dependent, then there exist constants not all zero satisfying the vector equation c1v1+c2v2+...+ckvk=0. (iii) Use this fact to derive a contradiction and conclude that T is linearly independent.arrow_forwardProof Prove that if S={v1,v2,,vn} is a basis for a vector space V and c is a nonzero scalar, then the set S1={cv1,cv2,,cvn} is also a basis for V.arrow_forward
- Proof Let {v1,v2,...,vn} be a linearly independent set of vectors in a vector space V. Delete the vector vk from this set and prove that the set {v1,v2,...,vk1} cannot span V.arrow_forwardProof Prove that if S1 and S2 are orthogonal subspaces of Rn, then their intersection consists of only the zero vector.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
- Guided Proof Prove that if u is orthogonal to v and w, then u is orthogonal to cv+dw for any scalars c and d. Getting Started: To prove that u is orthogonal to cv+dw, you need to show that the dot product of u and cv+dw is 0. i Rewrite the dot product of u and cv+dw as a linear combination of (uv) and (uw) using Properties 2 and 3 of Theorem 5.3. ii Use the fact that u is orthogonal to v and w, and the result of part i, to lead to the conclusion that u is orthogonal to cv+dw.arrow_forwardGuided Proof Let S be a spanning set for a finite dimensional vector space V. Prove that there exists a subset S of S that forms a basis for V. Getting Started: S is a spanning set, but it may not be a basis because it may be linearly dependent. You need to remove extra vectors so that a subset S is a spanning set and is also linearly independent. (i) If S is linearly independent set, then you are done. If not, remove some vector v from S that is a linear combination of the other vectors in S. Call this set S1. (ii) If S1 is a linearly independent set, then you are done. If not, then continue to remove dependent vectors until you produce a linearly independent subset S. (iii) Conclude that this subset is the minimal spanning set S.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
- True 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_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_forwardProof Use the properties of matrix multiplication to prove the first three properties of Theorem 5.3. Theorem 5.3 Properties of the Dot Product If u, v, and w are vectors in Rn and c is a scalar, then the properties listed below are true. 1. uv=vu 2. u(v+w)=uv+uw 3. c(uv)=(cu)v=u(cv)arrow_forward
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