# Linear Algebra - Subspace
## Definition
For all vector space,a subspace is a subset that is itself a vector space, under the inheriate operation.
## Lemma
If $W$ is a vector space, $V\subseteq W$, $\forall$ $\vec{v},\vec{w}\in V, r\in V$
1. $\vec{0}\in V$
2. $\vec{v}+\vec{w}\in V$
3. $r\vec{v}\in V$

,then V is subspace of W.
## Determine whether it is a subspace of a vector space
> $\mathbb{R}^2=span(\{\left[ {\begin{array}{} 1\\0 \end{array} } \right],\left[ {\begin{array}{} 0\\1 \end{array} } \right]\})$
### Case 1
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$V=\emptyset$ (empty set). Is set V a subspace of $\mathbb{R}^2$?
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Ans:
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V is not a subspace of $\mathbb{R}^2$ because $\vec{0}\notin V$, $V$ is not a vector sapce.
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### Case 2
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$V=\{\left[ {\begin{array}{} x\\y \end{array} } \right]
\mid x+y\le 1\}$. Is set V a subspace of $\mathbb{R}^2$?
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Ans:
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V is not a subspace of $\mathbb{R}^2$ because $\left[ {\begin{array}{} 1\\0 \end{array} } \right]+\left[ {\begin{array}{} 0\\1 \end{array} } \right]\notin V$, $V$ is not a vector sapce.
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### Case 3
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$V=\{\left[ {\begin{array}{} x\\y \end{array} } \right]
\mid x+y=0\}$. Is set V a subspace of $\mathbb{R}^2$?
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Ans:
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V is a subspace of $\mathbb{R}^2$ because $\vec{0}\in V$, $\vec{v}+\vec{w}\in V$, $r\vec{v}\in V$ and $V\in \mathbb{R}^2$
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### Case 4
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$V=\{\left[ {\begin{array}{} x\\y \end{array} } \right]
\mid x+y=1\}$. Is set V a subspace of $\mathbb{R}^2$?
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Ans:
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V is not a subspace of $\mathbb{R}^2$ because $\left[ {\begin{array}{} 0\\0 \end{array} } \right]\notin V$, $V$ is not a vector sapce.
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### Case 5
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$V=\{\left[ {\begin{array}{} x\\y \end{array} } \right]
\mid y>0\}$. Is set V a subspace of $\mathbb{R}^2$?
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Ans:
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V is not a subspace of $\mathbb{R}^2$ because$\left[ {\begin{array}{} 0\\0 \end{array} } \right]\notin V$, $V$ is not a vector sapce.
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