Let W={(x,y)∈R2 ∣ 2x+3y=0}. Let us show that W is a subspace of R2. Let (x1,y1),(x2,y2)∈W, a,b∈R. Then 2x1+3y1=0, 2x2+3y2=0. Taking into account that a(x1,y1)+b(x2,y2)=(ax1+bx2,ay1+by2) and 2(ax1+bx2)+3(ay1+by2)=2ax1+2bx2+3ay1+3by2=a(2x1+3y1)+b(2x2+3y2)=a⋅0+b⋅0=0, we conclude that a(x1,y1)+b(x2,y2)∈W, and thus W is a subspace of R2.
For any (x,y)∈W, x=−1.5y, and therefore, (x,y)=(−1.5y,y)=y(−1.5,1). It follows that the vector (−1.5,1) span W, and consequently, the dimension of W is 1.
The set v+W={v+w :w∈W} is called a coset of W. Let v=(a,b) be arbitrary element of R2. Denote by c=−2a−3b.Then v+W={(a,b)+(x,y) ∣(x,y)∈W}={(a+x,b+y) ∣2x+3y=0}={(x′,y′) ∣2(x′−a)+3(y′−b)=0}={(x′,y′) ∣2x′+3y′−2a−3b=0}={(x′,y′) ∣2x′+3y′+c=0}
Therefore, the cosets of W are lines 2x+3y+c=0 , where c∈R.
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