Consider the vector space R4 under usual addition and scalar multiplication. a) Determine whether the set S = {(2b − 3c, b, c, b - 5c): b, c are real numbers} forms a subspace in R4. b) Determine whether the set S = {(1,2,3,0),(-1, 1,8,1), (2,0, 1, 4), (1, 0, 8,1),(0,0,0,2)} forms a linearly independent set in Rª. Which vector(s) must be deleted so that the set is linearly independent. c) Determine whether the set S = {(1,−1,8, 5), (−2,0, 1, 0), (1, 1,0,0)} spans Rª. Which vector(s) must be added so that the set spans R¹. d) Find the dimension of the subspace spanned by the vectors (1,-1,0, 5), (2, 0,-1,3), (0, 2,-1,-7), (3, 1, -2, 1), (0, 4, 3,-1)}.
Consider the vector space R4 under usual addition and scalar multiplication. a) Determine whether the set S = {(2b − 3c, b, c, b - 5c): b, c are real numbers} forms a subspace in R4. b) Determine whether the set S = {(1,2,3,0),(-1, 1,8,1), (2,0, 1, 4), (1, 0, 8,1),(0,0,0,2)} forms a linearly independent set in Rª. Which vector(s) must be deleted so that the set is linearly independent. c) Determine whether the set S = {(1,−1,8, 5), (−2,0, 1, 0), (1, 1,0,0)} spans Rª. Which vector(s) must be added so that the set spans R¹. d) Find the dimension of the subspace spanned by the vectors (1,-1,0, 5), (2, 0,-1,3), (0, 2,-1,-7), (3, 1, -2, 1), (0, 4, 3,-1)}.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve this under usual addition and scalar multiplication
Transcribed Image Text:Consider the vector space R* under usual addition and scalar multiplication.
a) Determine whether the set S = {(2b – 3c, b, c, b – 5c ):b, c are real numbers}forms a
subspace in Rª.
b) Determine whether the set S = {(1, 2,3,0),(-1,1,8,1), (2,0,1,4), (1,0,8,1),(0,0,0,2)}
forms a linearly independent set in Rª. Which vector(s) must be deleted so that the set is
linearly independent.
c) Determine whether the set S = {(1,–1,8,5), (-2,0, 1, 0), (1, 1,0,0)} spans R4. Which
vector(s) must be added so that the set spans R*.
d) Find the dimens ion of the subspace spanned by the vectors
(1,–1,0,5),(2,0, – 1,3), (0,2, –1, -7),(3,1, –2,1), (0, 4,3,–1)}.
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