# Properties of subspace

### Properties of subspace

#### Lessons

A subspace of $\Bbb{R}^n$ is any set $S$ in $\Bbb{R}^n$ that has the three following properties:
1) The zero vector is in $S$
2) For each $u$ and $v$ in the set $S$, the sum of $u+v$ is in $S$ (closed under addition)
3) For each $u$ in the set $S$, the vector $cu$ is in $S$. (closed under scalar multiplication)
• Introduction
Properties of Subspace Overview:
a)
A Subspace in $\Bbb{R}^n$
• The zero vector
• Closed under scalar multiplication
• Example of a Subspace
• Example of not a Subspace

b)
Subspace of a span of vectors in $\Bbb{R}^n$
• Remember span = linear combination
• Showing a span of vectors is a subspace in $\Bbb{R}^n$

• 1.
Showing that a set is a subspace of $\Bbb{R}^n$
Is the following set a subspace of $\Bbb{R}^2$?

• 2.
Is the following set a subspace of $\Bbb{R}^2$?

• 3.
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 $\Bbb{R}^2$.

• 4.
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 $\Bbb{R}^2$.

• 5.
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 $\Bbb{R}^3$.

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5.
Subspace of $\Bbb{R}^n$
5.1
Properties of subspace
5.2
Column space
5.3
Null space
5.4
Dimension and rank