Quesrion 8 1) Suppose v1, V2, V3, V4 span a vector space V. Prove that the listV1-V2, V2 - V3, V3 – V4, V4 also spans V.[3] [ 21 [512) Find a number t such that 1,-3,19 is not linearly independent in R³.3) (a) Show that if we consider C a vector space orver R, then the list {1+ i,1- i} islinearly independent.(b) Show that if we consider Ca vector space orver C, then the list {1+i, 1- i} islinearly dependent.4) Prove or give a counterexample: If v1, v2,... Vm is a linearly independent list ofvectors in a vector space V (over either Q, R, C), then 5v, - 4v2, V2, .. Vmlinearly independent.5) Prove or give a counterexample: If v, vz,.. Vm and w1, W2, ... Wm are inearlyindependent lists of vectors in a vector space V (over either Q, R, C), then 5v1 +W1, V2 + W2, . Vm + Wm is linearly independent.6) Let u, v be vectors in the space V pver the field IF and c a scalar. Prove thatSpan(u, v) = Span(u, cu + v).7) Let u, v be vectors in the space V pver the field IF and c a non-zero scalar. Provethat Span(u, v) = Span(cu, v)8) Let u, v be non-zero vectors. Prove that (u, v) is linearly dependendent if andonly the vectors are scalar multiplies of one another.9) Let V be a vector space and assume that (v1, v2, V3) is a linearly independentsequence from V, w is a vector from V, and that (v1 + w, vz + w, v3+ w) is alinearly dependent. Prove that w e Span(v1, v2, V3).is...%3D

Answered: 8) Let u, v be non-zero vectors. Prove…

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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Quesrion 8

1) Suppose v1, V2, V3, V4 span a vector space V. Prove that the list
V1-V2, V2 - V3, V3 – V4, V4 also spans V.
[3] [ 21 [51
2) Find a number t such that 1,-3,19 is not linearly independent in R³.
3) (a) Show that if we consider C a vector space orver R, then the list {1+ i,1- i} is
linearly independent.
(b) Show that if we consider Ca vector space orver C, then the list {1+i, 1- i} is
linearly dependent.
4) Prove or give a counterexample: If v1, v2,... Vm is a linearly independent list of
vectors in a vector space V (over either Q, R, C), then 5v, - 4v2, V2, .. Vm
linearly independent.
5) Prove or give a counterexample: If v, vz,.. Vm and w1, W2, ... Wm are inearly
independent lists of vectors in a vector space V (over either Q, R, C), then 5v1 +
W1, V2 + W2, . Vm + Wm is linearly independent.
6) Let u, v be vectors in the space V pver the field IF and c a scalar. Prove that
Span(u, v) = Span(u, cu + v).
7) Let u, v be vectors in the space V pver the field IF and c a non-zero scalar. Prove
that Span(u, v) = Span(cu, v)
8) Let u, v be non-zero vectors. Prove that (u, v) is linearly dependendent if and
only the vectors are scalar multiplies of one another.
9) Let V be a vector space and assume that (v1, v2, V3) is a linearly independent
sequence from V, w is a vector from V, and that (v1 + w, vz + w, v3+ w) is a
linearly dependent. Prove that w e Span(v1, v2, V3).
is
...
%3D

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Since it's an if and only if proof, don't you have to prove "if linearly dependent, then scalar multiples" and then "if scalar multiple, then linearly dependent"?

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