Question

Let x be a vector in Rⁿ written as a column, and define U = {Ax:A∈M_mn}.

Show that U is a subspace of R^m.

Answer 1

Clearly the vector U is defined as the multiplication of matirix of order mxn by a matrix of order nx1 and so this results in a matrix of order mx1 and hence a vector in R_m

Now let U₁ , U₂ be two vectors in U such that :

U₁ = Ax and U₂ = Bx, where A and B are matrices in M_mn

Now U₁ + U₂ = Ax + Bx = (A+B)x (By distributive property of matrix addition when they are of same order)

Here A+B is again a matric of order mxn and hence in M_mn

So U₁ +U₂ is in U

Now consider a in R

Then aU₁ = a(Ax) = (aA)x

As a is a scalar and so aA is again a matrix in M_mn

Hence aU₁ is in U

So U is a subspace of R_m