Subspaces - Examples with Solutions

Definiton of Subspaces

If W is a subset of a vector space V and if W is itself a vector space under the inherited operations of addition and scalar multiplication from V, then W is called a subspace.1 , 2
To show that the W is a subspace of V, it is enough to show that

1. W is a subset of V
2. The zero vector of V is in W
3. For any vectors u and v in W, u + v is in W. (closure under additon)
4. For any vector u and scalar r, the product r · u is in W. (closure under scalar multiplication).

Examples of Subspaces

Example 1
   The set W of vectors of the form $$(x,0)$$ where $$x \in \mathbb{R}$$ is a subspace of $$\mathbb{R}^2$$ because:
W is a subset of $$\mathbb{R}^2$$ whose vectors are of the form $$(x,y)$$ where $$x \in \mathbb{R}$$ and $$y \in \mathbb{R}$$
The zero vector $$(0,0)$$ is in W
$$(x_1,0) + (x_2,0) = (x_1 + x_2 , 0)$$ , closure under addition
$$r \cdot (x,0) = (r x , 0)$$ , closure under scalar multiplication

Example 2
The set W of vectors of the form $$(x,y)$$ such that $$x \ge 0$$ and $$y \ge 0$$ is not a subspace of $$\mathbb{R}^2$$ because it is not closed under scalar multiplication.
Vector $$\textbf{u} = (2,2)$$ is in W but its negative $$-1(2,2) = (-2,-2)$$ is not in W.

Example 3
The set W of vectors of the form $$W = \{ (x,y,z) | x + y + z = 0 \}$$ is a subspace of $$\mathbb{R}^3$$ because
1) It is a subset of $$\mathbb{R}^3 = \{ (x,y,z) \}$$
2) The vector $$(0,0,0)$$ is in W since $$0 + 0 + 0 = 0$$
3) Let $$\textbf{u} = (x_1 , y_1 , z_1)$$ and $$\textbf{v} = (x_2 , y_2 , z_2)$$ be vectors in W. Hence
$$x_1 + y_1 + z_1 = 0$$ and $$x_2 + y_2 + z_2 = 0$$
$$(x_1 , y_1 , z_1) + (x_2 , y_2 , z_2) \\\\ \quad = (x_1+x_2 , y_1+y_2 , z_1+z_2) \\\\ \quad = (x_1+x_2) + (y_1+y_2) + (z_1+z_2) \\\\ \quad = (x_1+y_1+z_1) + (x_2+y_2+z_2) = 0 + 0 = 0$$
4) Let $$r$$ be a real number
$$r (x_1 , y_1 , z_1) = (r x_1 , r y_1 , r z_1)$$
$$r x_1 + r y_1 + r z_1 \\\\ \quad = r( x_1 + y_1 + z_1 ) \\\\ \quad = r \cdot 0 = 0$$
hence closure under scalar multiplication

1. Vector Spaces - Questions with Solutions
2. Linear Algebra and its Applications - 5 th Edition - David C. Lay , Steven R. Lay , Judi J. McDonald
3. Elementary Linear Algebra - 7 th Edition - Howard Anton and Chris Rorres