# Let M be a matrix and u and v vectors: M =[(a, b),(c, d)], v = [(x), (y)], u =[(w), (z)]. (a) Propose a definition for u + v. (b) Show that your definition obeys Mv + Mu = M(u + v)?

Jul 20, 2016

Definition of addition of vectors, multiplication of a matrix by a vector and proof of distributive law are below.

#### Explanation:

For two vectors $v = \left[\begin{matrix}x \\ y\end{matrix}\right]$ and $u = \left[\begin{matrix}w \\ z\end{matrix}\right]$
we define an operation of addition as $u + v = \left[\begin{matrix}x + w \\ y + z\end{matrix}\right]$

Multiplication of a matrix $M = \left[\begin{matrix}a & b \\ c & d\end{matrix}\right]$ by vector $v = \left[\begin{matrix}x \\ y\end{matrix}\right]$ is defined as $M \cdot v = \left[\begin{matrix}a & b \\ c & d\end{matrix}\right] \cdot \left[\begin{matrix}x \\ y\end{matrix}\right] = \left[\begin{matrix}a x + b y \\ c x + \mathrm{dy}\end{matrix}\right]$

Analogously, multiplication of a matrix $M = \left[\begin{matrix}a & b \\ c & d\end{matrix}\right]$ by vector $u = \left[\begin{matrix}w \\ z\end{matrix}\right]$ is defined as $M \cdot u = \left[\begin{matrix}a & b \\ c & d\end{matrix}\right] \cdot \left[\begin{matrix}w \\ z\end{matrix}\right] = \left[\begin{matrix}a w + b z \\ c w + \mathrm{dz}\end{matrix}\right]$

Let's check the distributive law of such definition:
$M \cdot v + M \cdot u = \left[\begin{matrix}a x + b y \\ c x + \mathrm{dy}\end{matrix}\right] + \left[\begin{matrix}a w + b z \\ c w + \mathrm{dz}\end{matrix}\right] =$

$= \left[\begin{matrix}a x + b y + a w + b z \\ c x + \mathrm{dy} + c w + \mathrm{dz}\end{matrix}\right] =$

=[(a(x+w)+b(y+z)),(c(x+w)+d(y+z)))]=

$= \left[\begin{matrix}a & b \\ c & d\end{matrix}\right] \cdot \left[\begin{matrix}x + w \\ y + z\end{matrix}\right] = M \cdot \left(v + u\right)$

End of proof.