Question

all vectors of the form (a,b,c) where b=a+c
is this subspace of R^3

Answer 1

A subspace S must satisfy the three conditions:
\[0\in S\]
\[\lambda u \in S\]
\[u +v \in S\]
when u en v are in S.

Answer 2

It is A subspace but it is obviously not "the" subspace.

Answer 3

Let's call your subspace S.
(0,0,0) satisfies 0=0+0, so the first condition is true.

when (a,b,c) is in S then b=a+c, then is also true: \[\lambda b=\lambda a+\lambda\]

So the second condition is satisfied

Answer 4

\[b _{u}=a _{u}+c _{u}\]
\[b _{v}=a _{v}+c _{v}\]
\[b _{v}+b _{u}=a _{v}+a _{u}+c _{v}+c _{u}\]

Which satisfies the third condition, so it is indeed a subspace of R^3

Answer 5

v = (a_1 + a_2, b_1 + b_2, c_1 + c_2)
b_1 = a_1 + c_1
b_2 = a_2 + c_2
b_1 + b_2 = a_1 + c_1 + a_2 + c_2
(b_1 + b_2) = (a_1 + a_2) + (c_1 + c_2)

Answer 6

Also it should be mentioned that the three mentioned properties are:
Existence of a zero vector,
Closure under addition
Closure under multiplication

Answer 7

i got the concept thanks sir