1. Label the following statements as true or false.(a) If S is a linearly dependent set, then each vector in S is a linearcombination of other vectors in S.(b)Any set containing the zero vector is linearly dependent.The empty set is linearly dependent.(c)(e)(d) Subsets of linearly dependent sets are linearly dependent.Subsets of linearly independent sets are linearly independent.If a₁1 + a₂x₂ + + anxn = 0 and 1,2,. ...,n are linearlyindependent, then all the scalars a are zero.(f)

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Answered: 1. Label the following statements as…

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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(e) Determine whether or not the rows of A are linearly independent. O independent O dependent (f) Let the columns of A be denoted by a,, az O (a, az, a} O (a, az, ag} O (a, ag, ag) and ag. Which of the following sets is (are) linearly independent? (Select all that apply.)
12 1 -1 1 0 2 2 -2 20 4 3 -3 3 0 6 Let A = -4 4 40-8 5 -5 5 0 10 6 -660 8 Which one of the following is NOT true about Col(A)? Select one alternative: Col(A) is a vector space of dimension 3 Col(A) is a subspace of R Col(A) is the span of Col(A) is the span of 123 -4 5 6 -2 2 -3 3 4 -5 5 2 3 -4 4 5 5 6 123 , which row reduces to 2 H N 246 -8 10 8 6 -8 10 8 [1 0 0 0 0 [0 -1 0 0 0 0 1 0 0 0 0 0 1 0 000 0 000 0 000
5. Determine (by justifying) whether the following set of vectors are linearly dependent or linearly independent. (a) { 1 } -2 -9- 2 (b) { 3 -1 7 14 2 1 (c) { 3 } -1 3 2
Mark each statement as TRUE/FALSE, Justify each answer.a) The set of all linear combinations of {v1, ... vp} is a vector space.b) Row operations on a matrix A can not change the linear dependence relations among the rows of A.
16. Show that if the non-zero vectors s), s2),..., s) are conjugate to a positive definite matrix G, then the vectors are independent.
3. Prove the following to establish theorem 7. (a) If the set of vectors {₁, 2, 3, ...} in R" are linearly dependent then at least one of the vectors can be expressed as a linear combination of the others. (b) If at least one of the vectors from the set containing vectors ₁, 2, 3, ... in Rn can be expressed as a linear combination of the others then the vectors form a linearly dependent set.
S is a linearly independent set,then for any x element of , x is not the linear combination of other vectors in S. Is this true or false?
is the ffollowing set of vectors in P2 is liner independen? Prove the answer

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1. Label the following statements as true or false. (a) If S is a linearly dependent set, then each vector in S is a linear combination of other vectors in S. (b) Any set containing the zero vector is linearly dependent. (c) The empty set is linearly dependent.