Short proofs. Carefully show your explanations. If u and v are solutions of the equation Ax = 0, show that where c is any real constant, is also a solution. If y is a solution of the equation Ax = b. and u is a solution of the equation Ax = 0, verify that is a solution of the equation . Solution 1) We know that Ax=0 and the solution of this equation is u and v, therefore we know that Au=0 So let us see if cu is also a solution A(cu) Because c is a scalar we can factor out that constant out in front. cAu Since we already know Au=0 c(Au) = 0 Therefore cu is also a solution. 2) So we know that y is a solution of Ax=b So we can write Ay=b Also we know that u is a solution of Ax=0 Au=0 We also know from the proof above that A(cu)=0 Now let us simply add the two equations Ay + A(cu) = b + 0 We can now factor out A A(y+cu) = b Hence y+cu is a solution of Ax=b as well..