Question

Let V=R^5 find two subspaces U and W such that
U⊕W=R^5

Answer 1

Answer on Question#40119 – Math – Linear Algebra:

Let $V = \mathbb{R}^5$. Find two subspaces $U, W$ such that $U \oplus W = \mathbb{R}^5$.

Solution.

Let $U = \{(a,0,0,0,0) \mid a \in \mathbb{R}\}$, $W = \{(0,b,c,d,e) \mid b,c,d,e \in \mathbb{R}\}$. Hence:

$$\begin{array}{l} u_1 = (x, 0, 0, 0, 0), u_2 = (y, 0, 0, 0, 0) \in U, \alpha \in \mathbb{R} \Rightarrow u_1 + \alpha u_2 = (x + \alpha y, 0, 0, 0, 0) \in U \Rightarrow \\ \Rightarrow U \text{ is a subspace}; \end{array}$$ $$\begin{array}{l} w_1 = (0, x, y, z, t), w_2 = (0, a, b, c, d) \in W, \alpha \in \mathbb{R} \Rightarrow \\ \Rightarrow w_1 + \alpha w_2 = (0, x + \alpha a, y + \alpha b, z + \alpha c, t + \alpha d) \in W \Rightarrow W \text{ is a subspace}; \end{array}$$ $$\forall v = (x_1, x_2, x_3, x_4, x_5) \in V: v = u + w,$$ $$\text{where } u \in U, w \in W, u = (x_1, 0, 0, 0, 0), w = (0, x_2, x_3, x_4, x_5);$$ $$\begin{array}{l} u_1, u_2 \in U, w_1, w_2 \in W, u_1 + w_1 = u_2 + w_2 \Rightarrow u_1 - u_2 = w_2 - w_1 \Rightarrow \\ \Rightarrow (a_1, 0, 0, 0, 0) - (a_2, 0, 0, 0, 0) = (0, b_2, c_2, d_2, e_2) - (0, b_1, c_1, d_1, e_1) \Rightarrow \\ \Rightarrow (a_1 - a_2, 0, 0, 0, 0) = (0, b_2 - b_1, c_2 - c_1, d_2 - d_1, e_2 - e_1) \Rightarrow a_1 - a_2 = 0 \Rightarrow u_1 = u_2 \Rightarrow \\ \Rightarrow w_1 = w_2; \end{array}$$

Hence:

$$\forall v \in V \exists! u \in U, w \in W: v = u + w \Rightarrow V = U \oplus W.$$