Vector Addition

<aside> 💡 Vector addition is a binary operation $+:V \times V \rightarrow V$, which takes any two vectors $v$ and $w$ of $V$ and assigns to them a third vector denoted by $v+w$.

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• In $\R ^n$, vector addition is defined by co-ordinate wise addition:

$(x_1,x_2,....,x_n) + (y_1,y_2,....,y_n)=(x_1+y_1,x_2+y_2,....,x_n+y_n)$

Scalar Multiplication

<aside> 💡 Scalar multiplication is a function $.:\R\times V\rightarrow V$, which take any element $c \in \R$ and $v\in V$ and assigns a new vector denoted by $c.v$.

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• In $\R ^n$, scalar multiplication is defined as follows:

$c(x_1,x_2,....,x_n)=(cx_1,cx_2,....,cx_n)$

Linear Combination of Vectors

Let $V$ be a vector space and $v_1,v_2,...,v_n \in V$:

<aside> 💡 The linear combination of $v_1,v_2,...,v_n$ with coefficients $a_1,a_2,....,a_n \in \R$ is the vector $\sum_{i=1}^na_iv_i$ $\in V$.

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<aside> 💡 A vector $v\in V$ is a linear combination of $v_1,v_2,...,v_n$ if there exists some $a_1,a_2,...,a_n$ $\in \R$ so that $v=\sum_{i=1}^na_iv_i$

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Definition of Linear Dependence

<aside> 💡 A set of vectors $v_1,v_2,...,v_n$ from a vector space $V$ is said to be linearly dependent, if there exists scalars $a_1,a_2,....,a_n$ , not all zero, such that: $a_1v_1+a_2v_2+....+a_nv_n=0$

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<aside> 💡 If a set is linearly dependent, then so is every superset of it.

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