Let A be an mn matrix and v and w vectors in Rn If Av0 and A

Let A be an m×n matrix and v and w vectors in R^n. If Av=0 and Aw=b, for any scalar c, show A(cv+w)=b.

Solution

Av=0 ----(1) (homogenous equation)

It can be written as : A(cv) =0 where c is any scalar

Aw = b ----(2)

Adding the above two A(cv) +Aw = A(cv +w) (Using the distributivity of matrix multiplication over addition) .

A(cv) +Aw = A(cv +w) = 0+b

A(cv +w) = cA(v) +Aw ( Associative property of matrix multiplication)

= c*0 +b

A(cv +w) = b