Vector Spaces and Subspaces - MIT Mathematics

1 Chapter 5 Vector Spaces and The Column Space of a MatrixTo a newcomer, matrix calculations involve a lot of you, they involve columns ofAvandABare linear combinations ofnvectors the columns ofA. Thischapter moves from numbers and vectors to a third level of understanding (the highestlevel). Instead of individual columns, we look at Spaces of vectors. Without seeingvector spacesand theirsubspaces, you haven t understood everything this chapter goes a little deeper, it may seem a little harder. That is natural. Weare looking inside the calculations, to find the Mathematics . The author s job is to make itclear. Section will present the Fundamental Theorem of Linear Algebra. We begin with the most important Vector Spaces . They are denoted byR1,R2,R3,R4,: : :. Each spaceRnconsists of a whole collection of all columnvectors with five components.

2 This is called 5-dimensionalspace. DEFINITIONThe spaceRnconsists of all column components ofvare real numbers, which is the reason for the letterR. When thencomponents are complex numbers,vlies in the Vector spaceR2is represented by the usualxyplane. Each vectorvinR2has twocomponents. The word space asks us to think of all those vectors the whole Vector gives thexandycoordinates of a point in the plane ; y/.Similarly the vectors inR3correspond to ; y; z/in three-dimensional one-dimensional spaceR1is a line (like thexaxis). As before, we print vectors as acolumn between brackets, or along a line using commas and parentheses : 4 is inR2; .1; 1; 0; 1; 1/is inR5; 1Ci1 i is inC2:The great thing about linear algebra is that it deals easily with five-dimensional don t draw the vectors, we just need the five numbers (ornnumbers).

3 251252 Chapter 5. Vector Spaces and SubspacesTo multiplyvby 7, multiply every component by 7. Here 7 is a scalar. To add vectorsinR5, add them a component at a time : five additions. The two essential Vector operationsgo oninside the Vector space, and they producelinear combinations:We can add any vectors inRn,and we can multiply any vectorvby any scalarc. Inside the Vector space means thatthe result stays in the space: This is inR4with components1; 0; 0; 1, then2vis the Vector inR4with components2; 0; 0; 2. (In this case 2 is the scalar.) A whole series of properties can be verified commutative law isvCwDwCv; the distributive law Vector space has a unique zero Vector satisfying0 CvDv. Those are three of theeight conditions listed in the Chapter 5 eight conditions are required of every Vector space. There are vectors other thancolumn vectors, and there are Vector Spaces other thanRn.

4 All Vector Spaces have to obeythe eight reasonable real Vector space is a set of vectors together with rules for Vector addition andmultiplication by real numbers. The addition and the multiplication must produce vectorsthat are in the space. And the eight conditions must be satisfied (which is usually noproblem). You need to see three Vector Spaces other thanRn:MYZThe Vector space ofall Vector space ofall Vector space that consists only of azero vectors are really matrices. InYthe vectors are functions oft, only addition is0C0D0. In each space we can add : matrices to matrices,functions to functions, zero Vector to zero Vector . We can multiply a matrix by4ora function by4or the zero Vector by4. The result is still spaceR4is four-dimensional, and so is the spaceMof2by2matrices. Vectorsin those Spaces are determined by four numbers.

5 The solutionspaceYis two-dimensional,because second order differential equations have two independent solutions. Section pin down those key words,independence of vectorsanddimension of a spaceZis zero-dimensional (by any reasonable definition of dimension). It is thesmallest possible Vector space. We hesitate to call itR0, which means no components you might think there was no Vector spaceZcontains exactly one space can do without that zero Vector . Each space has its own zero Vector thezero matrix, the zero function, the ; 0; 0 different times, we will ask you to think of matrices and functions as vectors. But at alltimes, the vectors that we need most are ordinary column vectors. They are vectors withncomponents butmaybe not allof the vectors withncomponents. There are importantvector spacesinsideRn. Those The Column Space of a ; 0; 0/PR3 LFigure : 4-dimensional matrix ofR3: planeP, lineL, with the usual three-dimensional spaceR3.

6 Choose a plane through the ; 0; 0/.That plane is a Vector space in its own we add two vectors in the plane,their sum is in the plane. If we multiply an in-plane Vector by2or 5, it is still in the plane in three-dimensional space is notR2(even if it looks likeR2/. The vectors havethree components and they belong toR3. The planePis a Vector illustrates one of the most fundamental ideas in linearalgebra. The plane ; 0; 0/is asubspaceof the full Vector a Vector space is a set of vectors (including0) that satisfiestwo requirements :Ifvandware vectors in the subspace andcis any scalar, then(i)vCwis in the subspaceand(ii)cvis in the other words, the set of vectors is closed under additionvCwand multiplicationcv(anddw). Those operations leave us in the subspace. We can also subtract, because wisin the subspace and its sum withvisv w.)

7 In short,all linear combinationscvCdwstay in the fact :Every subspace contains the zero Vector . The plane inR3has to go ; 0; 0/. We mention this separately, for extra emphasis, but it follows directly from rule (ii).ChoosecD0, and the rule requires0vto be in the that don t contain the origin fail those tests. Whenvis on such a plane, vand0varenoton the plane. A plane that misses the origin is not a through the origin are also Subspaces . When we multiply by 5, or add twovectors on the line, we stay on the line. But the line must go ; 0; 0/.Another subspace is all ofR3. The whole space is a subspace (of itself). That is afourth subspace in the figure. Here is a list of all the possible Subspaces ofR3:.L/Any line ; 0; 0/.R3/The whole plane ; 0; 0/ .Z/The single ; 0; 0/254 Chapter 5. Vector Spaces and SubspacesIf we try to keep onlypartof a plane or line, the requirements for a subspace don t at these examples 1 Keep only the ; y/whose components are positive or zero (this isa quarter-plane).

8 The ; 3/is included but. 2; 3/is not. So rule (ii) is violatedwhen we try to multiply bycD quarter-plane is not a 2 Include also the vectors whose components are both negative. Now we havetwo quarter-planes. Requirement (ii) is satisfied; we can multiply by anyc. But rule (i)now fails. The sum ; 3/andwD. 3; 2/is. 1; 1/, which is outside quarter-planes don t make a (i) and (ii) involve Vector additionvCwand multiplication by scalars likecandd. The rules can be combined into a single requirement the rule for Subspaces :A subspace containingvandwmust contain all linear 3 Inside the Vector spaceMof all 2 by 2 matrices, here are two Subspaces :.U/All upper triangular matrices a b0 d .D/All diagonal matrices a 00 d :Add any two matrices inU, and the sum is inU. Add diagonal matrices, and the sum isdiagonal. In this caseDis also a subspace ofU!

9 The zero matrix alone is also a subspace,whena,b, anddall equal a smaller subspace of diagonal matrices, we could requireaDd. The matrices aremultiples of the identity matrixI. TheseaIform a line of matrices the matrixIa subspace by itself ? Certainly not. Only the zero matrix mindwill invent more Subspaces of 2 by 2 matrices write them downfor Problem Column Space ofAThe most important Subspaces are tied directly to a matrixA. We are trying to solveAvDb. IfAis not invertible, the system is solvable for someband not solvable forotherb. We want to describe the good right sidesb the vectors thatcanbe written asAtimesv. Thoseb0sform the column space thatAvis a combination of the columns ofA. To get every possibleb, weuse every possiblev. Start with the columns ofA, andtake all their linear produces the column space contains not just thencolumns ofA!

10 DEFINITIONThe column space consists of all combinations of the columns:The combinations are all possible vectorsAv. They fill the column column space is crucial to the whole book, and here is solveAvDbisto expressbas a combination of the columns. The right sidebhas to be in the columnspaceproduced byAon the left side. Ifbis not ,AvDbhas no The Column Space of a Matrix255 The systemAvDbis solvable if and only ifbis in the column space in the column space, it is a combination of the columns. Thecoefficients in thatcombination give us a solutionvto the anmbynmatrix. Its columns havemcomponents (notn/. So thecolumns belong column space ofAis a subspace ofRm(notRn).The setof all column combinationsAxsatisfies rules (i) and (ii) for a subspace : When we addlinear combinations or multiply by scalars, we still produce combinations of the word subspace is always justifiedby taking all linear is a 3 by 2 matrixA, whose column space is a subspace ofR3.)