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Math Help - Linear Independence Proof

  1. #1
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    Linear Independence Proof

    I'm trying to finish these linear independence proofs:
    3. Let S = {v1, v2, v3} be a linearly independent subset of V and let
    T = {v1 + v2, v2 + v3, v1 + v3}.
    (a) Show that if char F is not 2, then T is linearly independent.
    (b) Show that if char F = 2, then T is not linearly independent.
    4. Show that if a subset S of V is linearly independent, then any nonempty subset T of S
    is also linearly independent.
    5. Show that if a subset S of V is linearly independent and v ∈ V is not in sp(S), then
    S ∪ {v} is linearly independent



    3. linear independent so a1v1+a2v2+a3v3=0 implies a1=a2=a3=0
    The characteristic is confusing me
    Like I want to say we have something like 1+1+....+1=0

    4.linear independent so a1v1+a2v2+a3v3=0 implies a1=a2=a3=0
    I know we want a1(v1+v2)+a2(v2+v3)+a3(v1+v3)=0 to imply a1=a2=a3=0
    we have a1v1+a1v2+a2v2+a2v3+a3v1+a3v3=0
    (a1v1+a2v2+a3v3)+a1v2+a2v3+a3v1=0
    a1v2+a2v3+a3v1=0
    5.linear independent so a1v1+a2v2+a3v3=0 implies a1=a2=a3=0
    v is not in sp(s), so not a linear combination
    so v is not in a1v1+a2v2+a3v3

    Any hints would be greatly appreciated
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  2. #2
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    3.a) suppose c1(v1+v2) + c2(v2+v3) + c3(v1+v3) = 0

    collecting terms, we have:

    (c1+c3)v1 + (c1+c2)v2 + (c2+c3)v3 = 0

    and so by the linear independence of {v1,v2,v3}:

    c1+c3 = 0
    c1+c2 = 0
    c2+c3 = 0

    hence c3 = c2 = -c1 (equations 1&2), so c2+c3 = -2c1 = 0

    now, IF the characteristic of F is NOT 2, then -2 ≠ 0, and we can divide by it (this is where we use the fact that char(F) ≠ 2).

    this gives us c1 = 0, and c2 = c3 follows immediately, which proves the linear independence of T.

    3.b) if you read the above carefully, you will see that if char(F) = 2, we can find non-zero elements c1,c2,c3 with c1(v1+v2) + c2(v2+v3) + c3(v1+v3) = 0

    (try some "simple" values for c1,c2,c3).

    4. this is very simple. if c1v1 + c2v2 + ...+ cnvn = 0 implies ALL the cj are 0,

    then suppose that our subset is the first k of the vj (by re-ordering them if needed).

    then we have c1v1 + c2v2 +...ckvk = 0, so

    c1v1 + c2v2 +...+ ckvk + 0v(k+1) +...+ 0vn = 0, which by the linear independence of the entire set means....?

    5. suppose we have c1v1 + c2v2 +...+ cnvn + dv = 0

    there are two cases: d = 0, and d ≠ 0.

    suppose d ≠ 0. then -dv = c1v1 + c2v2 +...+ cnvn, so

    v = (-c1/d)v1 + (-c2/d)v2 +...+ (-cn/d)vn, which means v is in span(S), but....(finish this)

    but if d = 0, what can we say about the remaining sum?
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