Question #151697
Let W = {(x, y) E R² | 2x + 3y = 0} Show
that W is a subspace of R2 . Find the
dimension of W. Also show that the cosets of
W are lines 2x + 3y + c = 0, where c belongs to R
1
Expert's answer
2020-12-21T17:09:40-0500

Let W={(x,y)R2  2x+3y=0}W = \{(x, y) \in \mathbb R^2\ |\ 2x + 3y = 0\}. Let us show that WW is a subspace of R2\mathbb R^2. Let (x1,y1),(x2,y2)W, a,bR.(x_1,y_1),(x_2,y_2)\in W,\ a,b\in \mathbb R. Then 2x1+3y1=0, 2x2+3y2=0.2x_1+3y_1=0, \ 2x_2+3y_2=0. Taking into account that a(x1,y1)+b(x2,y2)=(ax1+bx2,ay1+by2)a(x_1,y_1)+b(x_2,y_2)=(ax_1+bx_2, ay_1+by_2) and 2(ax1+bx2)+3(ay1+by2)=2ax1+2bx2+3ay1+3by2=a(2x1+3y1)+b(2x2+3y2)=a0+b0=0,2(ax_1+bx_2)+3(ay_1+by_2)=2ax_1+2bx_2+3ay_1+3by_2=a(2x_1+3y_1)+b(2x_2+3y_2)=a\cdot 0+b\cdot 0=0, we conclude that a(x1,y1)+b(x2,y2)Wa(x_1,y_1)+b(x_2,y_2)\in W, and thus WW is a subspace of R2.\mathbb R^2.


For any (x,y)W,(x, y)\in W, x=1.5yx=-1.5y, and therefore, (x,y)=(1.5y,y)=y(1.5,1)(x,y)=(-1.5y,y)=y(-1.5,1). It follows that the vector (1.5,1)(-1.5,1) span WW, and consequently, the dimension of WW is 1.


The set v+W={v+w :wW}v+W=\{v+w\ : w\in W\} is called a coset of WW. Let v=(a,b)v=(a,b) be arbitrary element of R2\mathbb R^2. Denote by c=2a3bc=-2a-3b.Then v+W={(a,b)+(x,y) (x,y)W}={(a+x,b+y) 2x+3y=0}={(x,y) 2(xa)+3(yb)=0}={(x,y) 2x+3y2a3b=0}={(x,y) 2x+3y+c=0}v+W=\{(a,b)+(x,y)\ | (x,y)\in 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 WW are lines 2x+3y+c=02x + 3y + c = 0 , where cR.c\in\mathbb R.




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