Solving Ax = 0 and Ax = bNull spacesDefinition 1. The null space of a matrix A isNul(A) = {x : Ax = 0}.In other words, if A is m × n, then it s null space consists of t hos e vectors x ∈ Rnwhich solve thehomogeneous e quationAx = 0.Theorem 2. If A is m × n, then Nul(A) is a subspac e of Rn.Proof. We need to show that Nul(A) is closed under addition and scalar multiplication: Solving Ax = 0 yields an explicit description of Nul(A).By that we mean a description as the span of some vectors.Example 3. F ind an explicit description of Nul(A) whe r eA =3 6 6 3 96 12 13 0 3.Solution.Note.The number of vectors in the spanning set for Nul(A) as derived above (w hichis as small as po ssible) equals the nu mber of free variables in Ax = 0.Armin [email protected] spacesDefinition 4. The column space Col(A) of a matri x A is the span of the columns of A.IfA = [a1an], then Col(A) = span{a1,, an}.• In other words, b is in Col(A) if and on ly if Ax = b has a solution.Why?• If A is m × n, then Col(A) is a subspace of Rm.Why?Example 5. F ind a matrix A such tha t W = Col(A) whereW =2x − y3y7x + y: x, y in R.Solution.Col(A) and solutions to Ax = bTheorem 6. Let xpbe a solution of th e equation Ax = b.Then every solution to Ax = b is of the form x = xp+ xn, w here xnis a so lution tothe homogeneo us equation Ax = 0.• In other words, { x : Ax = b} = xp+ Nul(A).• We often call xpa particular solution.The theorem then says that every solution toAx = b is the s um of a fixed chosen particularsolution and s ome soluti on toAx = 0.Proof. Let x be anot her solution to Ax = b. Then:Armin [email protected] Example 7. Le t A =1 3 3 22 6 9 7−1 −3 3 4and b =155.Using the RREF, find a paramet ric d e scription of the solu tions toAx = b:Every solution to Ax = b is therefore of the form:Note. A convenient way to just find a particular solution i s to set all free variables tozero.Practice problems• True or false?◦ The soluti ons to the equation Ax = b form a vec tor s pace.◦ The soluti ons to the equation Ax = 0 form a vec t or space.Armin [email protected] 8. F ind an explicit description of Nul(A) whe r eA =1 3 5 00 1 4 −2.Example 9. Is t h e given set W a vector space?If possible, expressW as the column or null space of some matrix A.(a) W =(xyz: 5x = y + 2z)(b) W =(xyz: 5x − 1 = y + 2z)(c) W =(xyx + y: x, y in R)Armin
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