4 FANG WANG Example 5.Find the RREF for the following matriz: 「10211b1 「10211 A-[A,= -1213-12 → 02 3 40b1+b2 0-21+b 00000 -b1+b2+b4 This tell us if there erists solution to Ar=b then vector b must satisfies the -2b1-b2+b3=0. (1) -b1+b2+b4=0. It is easy to check b satisfies (1)if and only if bE Ran(A).4 FANG WANG Example 5. Find the RREF for the following matrix: A˜ = [A, b] =     1 0 2 1 1 b1 −1 2 1 3 −1 b2 1 2 5 5 1 b3 2 −2 1 −2 2 b4     −→     1 0 2 1 1 b1 0 2 3 4 0 b1 + b2 0 2 3 4 0 −b1 + b3 0 −2 −3 −4 0 −2b1 + b4     −→     1 0 2 1 1 b1 0 1 3 2 2 0 1 2 (b1 + b2) 0 0 0 0 0 −2b1 − b2 + b3 0 0 0 0 0 −b1 + b2 + b4     This tell us if there exists solution to Ax = b then vector b must satisfies the (1) ( −2b1 − b2 + b3 = 0, −b1 + b2 + b4 = 0. It is easy to check b satisfies (1) if and only if b ∈ Ran(A)