Suppose I have the linear equationax+by=c\ \ \ (1)where (a,b,c)\in\mathbb(R)^

Myah Fuller

Myah Fuller

Answered question

2022-02-23

Suppose I have the linear equation
ax+by=c   (1)
where (a,b,c)R3. Let W={a,b,c)R3:(1) is consistent}.
Is W a subspace of R3?
Apparently the answer is no, but I am unable to find a counter-example. If anything, my reasoning (below) tells me W is a subspace.
1.(0,0,0)W.
2.If (a,b,c)W, then (αa,αb,αc)W for any non-zero αR
3. If (a,b,c),(a,b,c)W, then
(a+a,b+b,c+c)W

Answer & Explanation

Elijah Hunt

Elijah Hunt

Beginner2022-02-24Added 5 answers

(1,-1,0) and (-1,0,1) are consistent, but (1,-1,0)+(-1,0,1)=(0,0,1) is inconsistent.

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