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)#?

Question

Noah Atkinson · Accepted Answer

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

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.

Parker Allen · Answer

undefined (a) The definition of $ u + v $ for vectors $ u $ and $ v $ is:

$$ u + v = \begin{bmatrix} w + x \ z + y \end{bmatrix} $$

(b) To show that this definition obeys $ Mv + Mu = M(u + v) $, we first compute each side:

$$ Mv = M \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} ax + by \ cx + dy \end{bmatrix} $$

$$ Mu = M \begin{bmatrix} w \ z \end{bmatrix} = \begin{bmatrix} aw + bz \ cw + dz \end{bmatrix} $$

$$ u + v = \begin{bmatrix} w + x \ z + y \end{bmatrix} $$

Now, we compute $ M(u + v) $:

$$ M(u + v) = M \begin{bmatrix} w + x \ z + y \end{bmatrix} = \begin{bmatrix} a(w + x) + b(z + y) \ c(w + x) + d(z + y) \end{bmatrix} $$

Expanding, we get:

$$ = \begin{bmatrix} aw + ax + bz + by \ cw + cx + dz + dy \end{bmatrix} $$

$$ = \begin{bmatrix} aw + bz + (ax + by) \ cw + dz + (cx + dy) \end{bmatrix} $$

$$ = \begin{bmatrix} ax + by \ cx + dy \end{bmatrix} + \begin{bmatrix} aw + bz \ cw + dz \end{bmatrix} $$

$$ = Mv + Mu $$

Therefore, $ Mv + Mu = M(u + v) $, showing that the proposed definition obeys this equality.

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)#?