Let A be an m x n matrix and b a vector in Rn W x in Rn Ax

Let A be an m x n matrix and b a vector in Rⁿ

W = {x in Rⁿ | Ax = kb where k is any real number}

is W subspace of Rⁿ? Explain? (b is fixed but k can vary)

Solution

1. Let, x and y be in W

So there is some k1,k2 so that

Ax=k1b

Ay=k2b

A(x+y)=Ax+Ay=(k1+k2)b

Hence, x+y is in W

Let, c be a scalar

A(cx)=cAx=ck1b=(ck1)b

So, cx is in W

Hence W is closed under addition and scalar multiplication and hence a subspace of Rn