Question

How are the operations of addition, subtraction and scalar multiplication defined for vectors in $\mathbb{R}^n$?

Answer 1

If $\mathbf{v}=\left(v_{1}, v_{2}, \ldots, v_{n}\right)$ and $\mathbf{w}=\left(w_{1}, w_{2}, \ldots, w_{n}\right)$ are vectors in $R^{n}$, and if $k$ is any scalar, then we define $$ \begin{aligned} &\mathbf{v}+\mathbf{w}=\left(v_{1}+w_{1}, v_{2}+w_{2}, \ldots, v_{n}+w_{n}\right) \\ &k \mathbf{v}=\left(k v_{1}, k v_{2}, \ldots, k v_{n}\right) \\ &-\mathbf{v}=\left(-v_{1},-v_{2}, \ldots,-v_{n}\right) \\ &\mathbf{w}-\mathbf{v}=\mathbf{w}+(-\mathbf{v})=\left(w_{1}-v_{1}, w_{2}-v_{2}, \ldots, w_{n}-v_{n}\right) \end{aligned} $$