Problem 4 (Solution space of a homogeneous system) Let A be an mxn matrix. The goal of this prob- lem is to show that the set of solutions to the equation Ar=0 is a subspace of R", proving some properties of matrix-vector multiplication as we go. Write A = [aj], which is shorthand notation for and write = A 1 . a11 a12 ain a21 a22 a2n A = am1 am2 amn Recall that A is the vector in Rm whose i-th component is given by the formula a¿1x1 + ··· + ainn. In other words, 1 a11 a12 a1n a21 a22 a2n Ax= : In am1 am2 amn nx1 mxn a11x1 + + a1nxn am1x1++amnxn] mx1 1. Prove that for any matrix A Є Mmxn and any vector, ER", we have A(+ y) = A + Ay. 2. Prove that for any matrix AE Mmxn, any scalar c ER and any vector ЄR", we have A(cx) = c(A). 3. Prove that S = {ã € R^|Aï = 0} is a subspace of Rn.