<aside> 💡 If not able to view properly: Try this link: https://www.notion.so/kalyandath18/Week-3-047baf2997314d9c8451bad234ea0adf

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Properties of Addition and Scalar Multiplication

Let $v,w,v^{'}$ be vectors in $\R^n$ and $a,b \in \R$.Then:

1. $v+w=w+v$
2. $(v+w)+v^{'} =v+(w+v^{'})$
3. The $0$ vector satisfies that $v+0=0+v=v.$
4. The vector $-v$ satisfies that $v+(-v)=0$.
5. $1v=v$
6. $(ab)v=a(bv)$
7. $a(v+w)=av+aw$
8. $(a+b)v=av+bv$

Definition of a Vector Space

<aside> 💡 A vector space is a set with two operations (called addition and scalar multiplication with the above properties 1-8.

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<aside> 💡 A vector space $V$ over $\R$ is a set along with two functions: $+$$:V\times V \rightarrow V$ and $.~$$:\R \times V \rightarrow V$

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i.e., for each pair of elements $v_1$ and $v_2$ in $V$, there is a unique element $v_1+v_2$ in $V$, and for each $c \in \R$ and $v\in V$ there is a unique element $c.v$ in $V.$

<aside> 💡 The functions $+$ and $.$ are required to satisfy rules 1-8 of addition and scalar multiplication.

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Properties( Same as above )

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For $V$ to be a vector space, the following conditions should hold:

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