How Does Field Characteristic Affect Linear Independence?

[kathrynag]

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

 

[Mark44]

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

I don't know what F is or what char F is.

I think you might be missing an important point about linear independence. Namely, that if v1, v2, and v3 are linearly dependent, then the equation a1*v1 + a2*v2 + a3*v3 = 0 also has a solution for the scalars of a1 = a2 = a3 = 0. The fine point that many students miss is that if the vectors are linearly independent, then there is only one solution to the equation a1*v1 + a2*v2 + a3*v3 = 0. For linearly dependent vectors, there are also other solutions.

Since I don't know what F or char F are supposed to mean, I would show that the equation c1(v1 + v2) + c2(v2 + v3) + c3(v1 + v3) = 0 has exactly one solution for the three constants, given that v1, v2, and v3 are linearly independent.

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

 

[vela]

The characteristic of a field is the smallest positive number n where the sum of n 1's is equal to 0.

Let a(v₁+v₂)+b(v₂+v₃)+c(v₁+v₃)=0. Then (a+c)v₁+(a+b)v₂+(b+c)v₃=0. Since v₁, v₂, and v₃ are linearly independent, you must have

a+c = 0
a+b = 0
b+c = 0

Now try to solve those equations, taking into account char F.Based on your work, I think you've completely misunderstood problem 4. Can you elaborate on what you wrote? For both problems 4 and 5, a proof by contradiction is probably the way to go.