MATH : Matrices & Linear Algebra Spring Final Review

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1 MATH : Matrices & Liear Algebra Sprig 009 Fial Review Hua Sha Gauss-Jorda Eliiatio [.] Reduced row-echelo for (rref Rak [.3] rak(a = uber of leadig s i rref(a di(i A = rak( A Liear Trasforatio i Geoetry [.,.] Liear trasforatio: T :, T( x = Ax Scalig/Orthogoal proectio/reflectio/rotatio/shears Iverse of atrix [.3] Fie the reduced row-echelo for of a augeted atrix[ AI ]. Matrix ultiplicatio [.4] Properties of atrix ultiplicatio: A A. AC ( = ( AC A( + C = A+ AC If A= I, the = A. Cosider T :, T( x = Ax Iage: [3.] ef.: i( T = i( A = { y: y = Ax for all x } Properties: i( A is a subspace of (i( A cotais zero ad is closed uder liear cobiatio Kerel: [3.] ef.: ker( T = ker( A = { x : Ax = 0 } Properties: ker( A is a subspace of cobiatio Applicatios: Fid i( A ad ker( A. Fid diesios of i( A ad ker( A. Fid bases of i( A ad ker( A. ( ker( A cotais zero ad is closed uder liear

2 MATH : Matrices & Liear Algebra Sprig 009 Hua Sha Spa: [3.] Cosider v,, v. spa( v,, v = { cv + + c v, for all c,, c } Subspace of [3.] W ad W cotais the zero vector of ad W is closed uder liear cobiatio Liear idepedece: [3.] Vectors v, v are liearly idepedet if oe of the is redudat., asis of a subspace: [3.] Cosider v,, v. Vectors v,, v for a basis of a subspace V of V=spa( v,, v ad v,, v are liearly idepedet. Fid a basis of or a subspace of. Vectors v,, v for a basis of iff atrix [ v,, v ] is ivertible. Liear relatio: [3.] cv + + c v = 0 Trivial liear relatio: cv + + c v = 0 iff c = = c = 0 iesio: [3.3] Cosider a subspace V of. The basis of V is fored by vectors v, di( V = (uber of vectors i basis of V Cosider T :, T( x = Ax di(i A = rak( A di(ker A = rak( A ad di(i A + di(ker A = di( =., v if

3 MATH : Matrices & Liear Algebra Sprig 009 Hua Sha Suary Cosider a liear trasforatio T :, T( x = Ax. A is a trasforatio atrix with. The followig stateets are equivalet: The liear syste Ax = 0 has a uique solutio x = 0. rak( A = ( uber of free variables = rak( A = 0. ker( A = {0} There is oly the trivial relatio aog the colu vectors v,, v of atrix A (The colu vectors v,, v of atrix A are liearly idepedet.. di(ker A = rak( A = 0 (zero vector is redudat di(i A = rak( A = If =, the we ca add a few ore to the list A is ivertible rref ( A = I i( A = Coordiates: [3.4] Cosider vectors v,, v that for a basis of a subspace V of, i.e. = ( v,, v. The coordiate of ay vector x V is [ x] = [ c,, c ] T if x = cv + + c v or i atrix for c x [ v v ] = = S[ x] c Two types of questios: Sx [ ] = x or x Sx [ ] = atrix ad stadard atrix A: [3.4] = ( v,, v of. Cosider a liear trasforatio T :,. A is the stadard atrix of the trasforatio. ( T( x = Ax is the atrix of the trasforatio. ([ T( x ] = [ x] = [ T( v ] [ T( v ] Let S = [ v v ], we have AS S, A SS = =, = S AS (A ad are siilar atrices 3

4 MATH : Matrices & Liear Algebra Sprig 009 Hua Sha Subspace of a liear space [4.] A subset W of liear space V is a subspace of V if W cotais the eutral eleet of V ad W is closed uder liear cobiatio. iesio of a subspace: [4.] Cosider a subspace W of a liear space V. The basis of W is fored by eleets f,, f, the di( W = (uber of eleets i basis of W Orthooral vectors ad orthooral basis: [5.] Vectors,, u u are orthooral if they are all uit vectors ad orthogoal to each other. ui u = δi Vectors,, u u for a orthooral basis of a, i.e. U= ( u,, u. Orthogoal proectio o a subspace V: [5.] The orthogoal proectio of x oto a subspace V spaed by U= ( u,, u pro V ( x = x = x x = x ( u x u + + ( u x u [ ] Orthooral Copleet: [5.] Let V be a subspace of, the orthogoal copleet V is defied as V = { x : v x = 0 for all v V} Agle betwee two vectors: [5.] x y θ = cos x y 4

5 MATH : Matrices & Liear Algebra Hua Sha Sprig 009 Gra-Schidt Process: [5.] Vectors v,, v for a basis of a subspace V of, i.e. = ( v,, v. The Gra-Schidt process coverts = ( v,, v to U= ( u,, u pro V ( x = x = x x = x ( u x u + + ( u x u [ ] v v u = v v v = v v = v ( u v u v u = v v v = v v = v ( u v ( u + + u v u u = v v eteriat: [6., 6.3] atrices 3 3 atrices: Sarrus s Rule atrices: Laplace expasio dow the i-th row i+ det( A = ( a det( A for fixed i = Laplace expasio dow the -th colu i+ det( A = ( a det( A for fixed i= Geoetrical eaig of deteriat [6.3] I, Area of parallelogra fored by vectors i i v ad v = det( A = det[ v v] = v v I 3, Volue of parallelepiped fored by vectors v, v ad v 3 det( A = det[ vvv] = v v v = 3 3 i i 5

6 MATH : Matrices & Liear Algebra Sprig 009 Properties of eteriats: [6.] Hua Sha det( A = det( Adet( det( A = / det( A T det( A = det( A det( diag( ai = ai i= Suary Cosider a liear trasforatio T :, T( x = Ax. A is a trasforatio atrix. The followig stateets are equivalet: A is ivertible rref ( A = I det( A 0 rak( A = ( uber of free variables = rak( A = 0. The liear syste Ax = 0 has a uique solutio x = 0. ker( A = {0} i( A = There is oly the trivial relatio aog the colu vectors v,, v of atrix A (The colu vectors,, v v of atrix A are liearly idepedet..,, v v for a basis of di(ker A = rak( A = 0 (zero vector is redudat di(i A = rak( A = Orthogoal Matrix: [6.3] A atrix Q is a orthogoal atrix if det( Q =± T QQ= I. Craer s Rule: [6.3] Ax= b. A is a ivertible atrix ( det( A 0 det( A bi, xi = i =,, det( A 6

7 MATH : Matrices & Liear Algebra Sprig 009 Hua Sha Eige Proble: [7., 7., 7.3] Cosider a atrix A. 0 v that solves A x = λ x for soe scalar λ is called the eigevector of A. λ is called the eigevalue associated with the eigevector. If v is a eigevector of atrix A, the v is a eigevector of atrices 3 3 Av = λ v, Av = λ v, 3 A, A, with Characteristic Polyoial f ( λ = det( A λi Fid Eigevalues ad Eigevectors: [7., 7.3] Eigevalues: Solve fa( λ = det( A λi = 0 Algebraic ultiplicity of eigevalue Eigevectors: Solve Ax= λ x or ( A λi x = 0 Eigespace ad Eigebasis: [7.3] A Eigespace: Eλ = ker( A λi Geoetric ultiplicity of eigevalue λ = di( Eλ = di(ker( A λi = rak( A λi Eigebasis: A basis of that cosists of eigevectors ( v, v of A. To fid eigespace ad eigebasis Solve fa( λ = det( A λi = 0. Fid E = ker( A λi spaed by eigevectors ad di( E λ. λ 3 Copute s = di( ad copare with : E λ If s =, the the eigebasis exists ad it cosists of all the eigevectors foud i step. If s <, the the eigebasis does ot exist. ut still, the eigevectors foud i step are liearly idepedet., 7

8 MATH : Matrices & Liear Algebra Sprig 009 iagoalizatio: [7.4] Hua Sha Cosider a liear trasforatio T :, T( x = Ax. A is a trasforatio atrix. Suppose the eigebasis for A exists: = ( v,, v. The coordiate of ay vector x V is [ x] = [ c,, c ] T if x = cv + + c v or i atrix for c x [ v v ] = = S[ x] c where S = [ v v ] Let be the atrix of the trasforatio. ([ T( x ] = [ x] As we leared fro sectio 3.4, that = [ T( v ] [ T( v ] ad atrices A ad are siilar, i.e. AS = S, = S AS ecause v, v are eigevectors of atrix A, we ca show that atrix is diagoal., A atrix A is diagoalizable if ad oly if there exists a eigebasis for A. If a atrix A has distict eigevalues, the A is diagoalizable., Algorith - iagoalizatio: [7.4] Solve f ( λ = det( A λi = 0. A Fid E = ker( A λi spaed by eigevectors ad di( E λ. λ 3 Copute s = di( ad copare with : E λ If s =, the the eigebasis exists ad it cosists of all the eigevectors foud i step. Let S = [ v v ] ad copute = S AS. If s <, the the eigebasis does ot exist thus atrix A is ot diagoalizable. 8

) is a square matrix with the property that for any m n matrix A, the product AI equals A. The identity matrix has a ii

) is a square matrix with the property that for any m n matrix A, the product AI equals A. The identity matrix has a ii square atrix is oe that has the sae uber of rows as colus; that is, a atrix. he idetity atrix (deoted by I, I, or [] I ) is a square atrix with the property that for ay atrix, the product I equals. he