vectors form a basis of the R³ vector space?0.074ectors y₁, Y2, Y3 (where y; E R² Vi) must be linearly dependent?345

ℝ3 vector space is a three dimensional space. So, the basis for ℝ3 must have exactly three linearly…

Answered: vectors form a basis of the R³ vector…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.3: Orthonormal Bases:gram-schmidt Process

Problem 63E

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Proof 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.
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 property
Guided 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.
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.
Which vector spaces are isomorphic to R6? a M2,3 b P6 c C[0,6] d M6,1 e P5 f C[3,3] g {(x1,x2,x3,0,x5,x6,x7):xiisarealnumber}
Find a basis for R2 that includes the vector (2,2).

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vectors form a basis of the R³ vector space? (-)-() 7 4 4 5 ctors y₁, y₂, Y3 (where y; E R² Vi) must be linearly dependent?