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23–24 Determine whether the given
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Chapter 12 Solutions
Calculus (MindTap Course List)
- Illustrate properties 110 of Theorem 4.2 for u=(2,1,3,6), v=(1,4,0,1), w=(3,0,2,0), c=5, and d=2. THEOREM 4.2Properties of Vector Addition and Scalar Multiplication in Rn. Let u,v, and w be vectors in Rn, and let c and d be scalars. 1. u+v is vector in Rn. Closure under addition 2. u+v=v+u Commutative property of addition 3. (u+v)+w=u+(v+w) Associative property of addition 4. u+0=u Additive identity property 5. u+(u)=0 Additive inverse property 6. cu is a vector in Rn. Closure under scalar multiplication 7. c(u+v)=cu+cv Distributive property 8. (c+d)u=cu+du Distributive property 9. c(du)=(cd)u Associative property of multiplication 10. 1(u)=u Multiplicative identity propertyarrow_forwardProof 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_forwardTake this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. a Explain what it means to say that a set of vectors is linearly independent. b Determine whether the set S is linearly dependent or independent. S={(1,0,1,0),(0,3,0,1),(1,1,2,2),(3,4,1,2)}arrow_forward
- A vector in three dimensions can be written in either of two forms: in coordinate form as v=a1,a2,a3 and in terms of the _________ vectors i,j, and k as v= __________. The magnitude of the vector v is |v|= _________. So 4,2,4=i+j+k and 7j24k=,,.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_forward
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