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The problem is attached.

I'm having problems with parts a and c, well maybe not part a (probably just need to check if I did this part right. I'm just not sure if I'm wording part a right.

Anyways for part a I must prove it's a subspace so I must satisfy 3 conditions:

1) 0 is in S

2) if U and V are in S, then U+V must be in S

3) if V is in S, then fV is in S for some scalar f.

for 1)

By inspection if a=b=c=0 then 0 is in S

for 2)

if U is of the form:

a1-b1 a1

b1+c1 a1-c1

and V is of the form:

a2-b2 a2

b2+c2 a2-c2

then U+V=

a1+a2-b1-b2 a1+a2

b1+b2+c1+c2 a1+a2-c1-c2

Thus U+V is in S. <<< Can I say this?

for 3)

fV=

f(a2-b2) f(a2)

f(b2+c2) f(a2-c2)

Thus fV is in S. <<< Can I say this?

For part c, I don't even know where to begin. Can someone give me a hint?

I'm having problems with parts a and c, well maybe not part a (probably just need to check if I did this part right. I'm just not sure if I'm wording part a right.

Anyways for part a I must prove it's a subspace so I must satisfy 3 conditions:

1) 0 is in S

2) if U and V are in S, then U+V must be in S

3) if V is in S, then fV is in S for some scalar f.

for 1)

By inspection if a=b=c=0 then 0 is in S

for 2)

if U is of the form:

a1-b1 a1

b1+c1 a1-c1

and V is of the form:

a2-b2 a2

b2+c2 a2-c2

then U+V=

a1+a2-b1-b2 a1+a2

b1+b2+c1+c2 a1+a2-c1-c2

Thus U+V is in S. <<< Can I say this?

for 3)

fV=

f(a2-b2) f(a2)

f(b2+c2) f(a2-c2)

Thus fV is in S. <<< Can I say this?

For part c, I don't even know where to begin. Can someone give me a hint?

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