Question

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)`?

Answer 1

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

Explanation:

For two vectors `v=[(x),(y)]` and `u=[(w),(z)]`
we define an operation of addition as `u+v=[(x+w),(y+z)]`

Multiplication of a matrix `M=[(a,b),(c,d)]` by vector `v=[(x),(y)]` is defined as `M*v =[(a,b),(c,d)]*[(x),(y)] = [(ax+by),(cx+dy)]`

Analogously, multiplication of a matrix `M=[(a,b),(c,d)]` by vector `u=[(w),(z)]` is defined as `M*u =[(a,b),(c,d)]*[(w),(z)] = [(aw+bz),(cw+dz)]`

Let's check the distributive law of such definition:
`M*v+M*u= [(ax+by),(cx+dy)]+[(aw+bz),(cw+dz)]=`

`=[(ax+by+aw+bz),(cx+dy+cw+dz)]=`

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

`= [(a,b),(c,d)] * [(x+w),(y+z)] = M*(v+u)`

End of proof.