# Properties of subspace

##### Intros
###### Lessons
1. Properties of Subspace Overview:
2. A Subspace in $\Bbb{R}^n$
• The zero vector
• Closed under scalar multiplication
• Example of a Subspace
• Example of not a Subspace
3. Subspace of a span of vectors in $\Bbb{R}^n$
• Remember span = linear combination
• Showing a span of vectors is a subspace in $\Bbb{R}^n$
##### Examples
###### Lessons
1. Showing that a set is a subspace of $\Bbb{R}^n$
Is the following set a subspace of $\Bbb{R}^2$?
1. Is the following set a subspace of $\Bbb{R}^2$?
1. Showing that a set is a subspace of $\Bbb{R}^n$ with graphs
The following graph displays a set in $\Bbb{R}^2$. Assume the set includes the bounding lines. Give a reason as to why the set $S$ is not a subspace of $\Bbb{R}^2$.
1. Showing that a set is a subspace of $\Bbb{R}^n$ with graphs
The following graph displays a set in $\Bbb{R}^2$. Assume the set includes the bounding lines. Give a reason as to why the set $S$ is not a subspace of $\Bbb{R}^2$.
1. Showing a set equal to a span of vectors is a subspace of $\Bbb{R}^n$
Let $U=$ Span{$v_1,v_2,v_3$}, where $U$ is a set. Determine if $U$ is in the subspace of $\Bbb{R}^3$.