Scalar by Vector Multiplication

A scalar by vector multiplication is also defined by multiplying the vector entry by entry.

If

$\alpha\in\mathbb{R}$

and

$\vec{x}=\begin{bmatrix} a_1\\ a_2\\ a_3\\ :\\a_n\end{bmatrix}\in\mathbb{R^n}$

then

$\vec{}y=\alpha\vec{x}=\begin{bmatrix}\alpha a_1\\\alpha a_2\\ \alpha a_3\\ :\\\alpha a_n\end{bmatrix}\in\mathbb{R^n}$

Equation 5

(Notice that each element of the vector is multiplied by the same scalar alpha).

Ask yourself, what happens graphically to the vector when multiplied by a scalar?

We will explore this in the next quiz.