Question #219111

Let W = {(X1, X2, X3) €R³: X2 + X3 = 0}. How do I show that W is a subspace of R³? What are two subspaces W1 and W2 of R³ such that R³=W⊕W1and R³=W⊕W2 but W1≠W2?

1
Expert's answer
2021-07-21T13:14:57-0400

Let p,qRp,q\in \mathbb{R} and (a1,a2,a3),(b1,b2,b3)W(a_1,a_2,a_3),(b_1,b_2,b_3)\in W.

Consider

p(a1,a2,a3)+q(b1,b2,b3)=(pa1+qb1,pa2+qb2,pa3+qb3)p(a_1,a_2,a_3)+q(b_1,b_2,b_3)\\=(pa_1+qb_1,pa_2+qb_2,pa_3+qb_3)

Now

pa2+qb2+pa3+qb3=p(a2+a3)+q(b2+b3)=0pa_2+qb_2+pa_3+qb_3\\=p(a_2+a_3)+q(b_2+b_3)=0

So p(a1,a2,a3)+q(b1,b2,b3)Wp(a_1,a_2,a_3)+q(b_1,b_2,b_3)\in W, and hence WW is a subspace of R3\mathbb{R}^3.


Next, let W1={(x2,x2,0)}W_1=\{(x_2,x_2,0)\} and W2={(x2,2x2,0)}W_2=\{(x_2,2x_2,0)\}.

Suppose

sWW1\textbf{s}\in W\cap W_1

Then every element of s\textbf{s} is of the form (x1,x2,x3)(x_1,x_2,x_3) where x3=0x_3=0, x2=x3=0x_2=-x_3=0 and x1=x2=0x_1=x_2=0. Hence s={(0,0,0)}\textbf{s}=\{(0,0,0)\}.

Also, if (a,b,c)R3(a,b,c)\in \mathbb{R}^3 then it can be written as

(a,b,c)=(abc,c,c)+(b+c,b+c,0)(a,b,c)=(a-b-c,-c,c)+(b+c,b+c,0)

So R3=WW1\mathbb{R}^3=W\bigoplus W_1.

Similarly, suppose tWW2\textbf{t}\in W \cap W_2. Then every element of t\textbf{t} is of the form (y1,y2,y3)(y_1,y_2,y_3) where y3=0y_3=0, y2=y3=0y_2=-y_3=0 and y1=y2=0y_1=y_2=0.

Also, if (a,b,c)R3(a,b,c)\in \mathbb{R}^3 then it can be written as

(a,b,c)=(2abc2,c,c)+(b+c2,b+c,0)(a,b,c)=\left(\frac{2a-b-c}{2},-c,c\right)+\left(\frac{b+c}{2},b+c,0\right)

So R3=WW2\mathbb{R}^3=W\bigoplus W_2.

Clearly, W1W2W_1\neq W_2.


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