arrow_back If $\mathbf{v}$ is a vector in $R^{n}$, and if $k$ is any scalar, then prove that $\|k \mathbf{v}\|=|k|\|\mathbf{v}\|$

Question

If $\mathbf{v}$ is a vector in $R^{n}$, and if $k$ is any scalar, then prove that $\|k \mathbf{v}\|=|k|\|\mathbf{v}\|$

Answer 1

If $\mathbf{v}=\left(v_{1}, v_{2}, \ldots, v_{n}\right)$, then $k \mathbf{v}=\left(k v_{1}, k v_{2}, \ldots, k v_{n}\right)$, so $\|k \mathbf{v}\|=\sqrt{\left(k v_{1}\right)^{2}+\left(k v_{2}\right)^{2}+\cdots+\left(k v_{n}\right)^{2}}$ $$ \begin{aligned} &=\sqrt{\left(k^{2}\right)\left(v_{1}^{2}+v_{2}^{2}+\cdots+v_{n}^{2}\right)} \\ &=|k| \sqrt{v_{1}^{2}+v_{2}^{2}+\cdots+v_{n}^{2}} \\ &=|k|\|\mathbf{v}\| \end{aligned} $$