Answer to Question #219111 in Linear Algebra for Dhananjay

Question

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?

Answer 1

Let $p,q\in \mathbb{R}$ and $(a_1,a_2,a_3),(b_1,b_2,b_3)\in W$.

Consider

$$p(a_1,a_2,a_3)+q(b_1,b_2,b_3)\\=(pa_1+qb_1,pa_2+qb_2,pa_3+qb_3)$$

Now

$$pa_2+qb_2+pa_3+qb_3\\=p(a_2+a_3)+q(b_2+b_3)=0$$

So $p(a_1,a_2,a_3)+q(b_1,b_2,b_3)\in W$, and hence $W$ is a subspace of $\mathbb{R}^3$.

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

Suppose

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

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

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

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

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

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

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

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

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

Clearly, $W_1\neq W_2$.