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content/Math429/Math429_L2.md

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+# Lecture 2
+
+## Chapter I Vector Spaces
+
+### Subspaces 1C
+
+#### Definition 1.33
+
+A subset $U$ of $V$ is called subspace of $V$ is $U$ is also a vector space with the same additive identity, addition and scalar multiplication as on $V$.
+
+#### Theorem 1.34
+
+Condition for a subspace.
+
+* Additive identity: $0\in U$
+* Closure under addition: $\forall u,w\in U,u+w\in V$
+* Closure under **scalar multiplication**: $a\in \mathbb{F}$ and $u\in V$, $a\cdot u\in V$
+
+Proof If $U$ is a subspace of $V$, then $U$ satisfies the three conditions above by the definition of vector space.
+
+Conversely, suppose $U$ satisfies the three conditions above. The first condition ensures that the additive identity of
+$V$ is in $U$. 
+
+The second condition ensures that addition makes sense on $U$. The third condition ensures that scalar multiplication makes sense on $U$.
+
+If $u\in U$, then $-u$ is also in $U$ by the third condition above. Hence every element of $U$ has an additive inverse in $U$. The other parts of the definition of a vector space, such as associativity and commutativity, are automatically satisfied for $U$ because they hold on the larger space $V$. Thus $U$ is a vector space and hence is a subspace of $V$.
+
+### Definition 1.36
+
+Sum of subspaces
+
+Suppose $V_1,...,V_m$ are subspace of $V$. The sum of $V_1,...,V_m$, denoted by $V_1+...+V_m$ is the set of all possible sum of elements of $V_1,...,V_m$.
+
+$$
+V_1+...+V_m=\{v_1+...+v_m:v_1\in V_1, ..., v_m\in V_m\}
+$$
+
+Example
+
+a sum of subspaces of $\mathbb{F}^3$
+
+Suppose $U$ is the set of all elements of $\mathbb{F}^3$ whose second and third coordinates equal 0, and 𝑊 is the set of all elements of $\mathbb{F}^3$ whose first and third coordinates equal 0:
+
+$$
+U = \{(x,0,0) \in \mathbb{F}^3 : x\in \mathbb{F}\} \textup{ and } W = \{(0,y,0) \in \mathbb{F}^3 :y\in  \mathbb{F}\}.
+$$
+
+Then
+
+$$
+U+W= \{(x,y,0)  \in \mathbb{F}^3 : x,y \in \mathbb{F}\}
+$$