# Subspaces - Examples with Solutions

   

## Definiiton 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 $\textbf{u}$ and $\textbf{v}$ in W, $\textbf{u} + \textbf{v}$ is in W. (closure under additon)
4. For any vector $\textbf{u}$ and scalar $r$, $r \cdot \textbf{u}$ 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
$r (x_1 , y_1 , z_1) = (r x_1 , r y_1 , r z_1)$
$r x_1 + r y_1 + r z_1 \\\\ \quad = r( x_1 + y_1 + z_1 ) \\\\ \quad = r \cdot 0 = 0$