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## LINEAR ALGEBRA

by: Maverick Koelpin

24

0

9

# LINEAR ALGEBRA MATH 2085

Maverick Koelpin
LSU
GPA 3.77

Staff

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COURSE
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Staff
TYPE
Class Notes
PAGES
9
WORDS
KARMA
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## Popular in Mathematics (M)

This 9 page Class Notes was uploaded by Maverick Koelpin on Tuesday October 13, 2015. The Class Notes belongs to MATH 2085 at Louisiana State University taught by Staff in Fall. Since its upload, it has received 24 views. For similar materials see /class/222641/math-2085-louisiana-state-university in Mathematics (M) at Louisiana State University.

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Date Created: 10/13/15
Supplemental Notes for Final Chapter 2 1 Afunction T Rm gtR Is calle a linear Transformation is a TvwTvTw b TkvkTv 2 A function TRm gtRquot is a linear equation iff there exist an nxm matrix A such that TxAx for all XCR39 Proof that they are equivalent TxAx TvwAvwAvAw TvTwAvAwAvw TkvAkvkAv kTvkAvkAvAkv Section 31 a Let TR gtRquot be a linear transformation then the kernel of T kerT is the set of all XCR39 such that Tx0 KerTx Rm Tx0 The image of a linear transformation TxAx is the span of the linearly independent column vectors of AmTb Rquot Txb We say that V1vspansRquot if every v in Rquot can be expressed as a linear combination of v1 v 57 0 Section 32 A subset W of vector space Rquot is called a subspace of Rquot if it has the followthree properties a W contains the zero vector b W is closed under addition c W is closed under scalar multiplication n Consider vectors vvm In R a We say that v is redundant if v is a linear combination of vectors V1Vi1 b The vectors V1Vm are called linearly independent if none of them is redundant Otherwise they are called linearly dependent at least one is redundant c We say that vectors V1Vm form a basis of a subspace V of Rquot if they span V and are linearly independent Consider the vectors V1Vm in Rquot An equation of the form c1v1cmvm0 is called a linear relation among the vectors There is always the trivial relation with c1c2cm0 Nontrivial relations where at least one coefficient c is nonzero may or may not exist among the vectors If TAx is a linear transformation from Rm to Rquot then 0 KerTkerA o lmTimA To construct a basis of the image of a matrix A list all the column vectors of A and omit the redundant vectors from this list Li gig 12 r M j 139 a spam i Consider vectros V1Vm in Rquot If v1 is nonzero and if each of the vectors v for i22 has a nonzero entry in a component where all the preceding vectors V1Vi1 have a 0 then the vectors V1Vm are linearly independent rln39 in The vectors V1Vm in Rquot are linearly dependent iff there are nontrivial relations among them 71 Tim i39 39i r The vectors in the kernel of an nxm matrix A correspond to the linear relations among the column vectors V1Vm of A The equation AxO means that x1v1xmvm0 f L Ii 1 ii f v Consider the vectors V1Vm in a subspace V of Rquot The vectors form a basis of V iff every vector v can be uniquely expressed as a linear combination vc1v1cmvm Section 34 Consider the basis Bv1vm of a subspace V of Rquot By the Basis and Unique Representation Theorem xc1v1cmvm The scalars c1cm are called the BCoordinates of x and the vectors c1cm is the Bcoordinate vector of x denoted by xB Also xSxB Where Sis a nxm Sv1 vm j39iwi 39 v Consider a linear transformation T from Rquot to Rquot and a basis B of Rquot The nxn matrix B that transforms xB to TxB for all x in Rquot We can construt B column by column as follows If BV1Vm then BTV1B TVmB If B is a basis of a subspace V of Rquot then 3 ixyiaixiaiyia b kXBkXB Consider a linear transformation T from Rquot to Rquot and a basis Bv1vm of Rquot Let B be the B matrix of T and let A be the standard matrix of T such that TxAx for all x in Rquot Then ASSB where Sv1 vn Section 41 A linear Space V is a set endowed with a rule for addition iff and g are in V then so is fg and a rule for scalar multiplication iff is in V and k is in R then kf is in V such that these operation satisfy the following eight rules for all f g h in V and all c k in R f9hf9h fggf There exist a neutral element n in V such that fnf This n is unique and denoted by O For each f in V there exist a g in V such that fg0 This 9 is unique and denoted by PPM f kfgkfkg ckfcfkf ckfckf 1ff 9 9 9 A subset W of a linear space V is called a subspace of V if a Wcontains the neutral element 0 of V b W is closed under addition if f and g are in W then so is fg c W is closed under scalar multiplication iff is in Wand k is in R then kf is in W var Unit y it v v4 1 Consider the elements f1f in a linear space V a We say that f1 fspans V if everyf in V can be expressed as a linear combination of f1f b We say that f is redundant if it is a linear combination of f1fi1 The elements f1f are called linearly independent if none of them is redundant c We say that elements f1f are a basis ofV if they span V and are linearly independent This means that everyf in V can be written uniquely as a linear equation fc1f1cf The coefficients c1c are called the coordinates of f with respect to the basis Bf1f d The vector c1 c2c is the Bcoordinate vector Write down generic element of space V in terms of arbitrary constants a b Using the arbitrary constant as coefficients express elements as a linear combination of some elements of V c Check if elements span and are Ll If a linear space V has a basis with n elements then all their bases of V consist of n elements as well We say that n is the dimension of V dimVn Section 42 d Let TR39 gtRquot be a linear transformation then the kernel of T kerT is the set of all XCR39 such that Tx0 e The image of a linear transformation TxAx is the span of the linearly independent column vectors of A DVranTnllityTdimimTdimkerT V in An invertible linear transformation T is called an isomorphism We say that the linear space V is isomorphic to the linear space W if there exists an isomorphism T from V to W Nquot mirv Let V be a vector space of dimension n So V has a basis let s denoted it by Bf1f To show that V is isometric to Rquot we need to showthere exists a linear transformation from Vto Rquot that is invertible Consider LBVgtRquot such that LBffB LB is linear by linearity of coordinates theorem It remains to showthat LB is invertible We claim that LB391RquotgtV is LB391k1k LBoLB391k1knLBk1f1kfk1kgtdR LB391oLBfLB391oLBt1f1tfLB391t1tnt1f1tffgtdv Thus LB is an isomorphism so V is isometric Rquot Section 51 Orthogonality length unit vector Two vectors v and w in Rquot are called orthogonal if vow0 The length of a vectorv in Rquot is vsqrtvov Avector u in Rquot is call unit vector if its length is 1 Orthonormal vectors The vectors u1um in Rquot are called orthonormal if they are all unit vector and orthogonal to one another uou1 if ij O i j Properties of orthonormal vectors 0 Orthonormal vectors are linearly independent 0 Consider a relation c1u1cucmum0 among the orthnormal vectors u1um in Rquot Let us form the dot product on each side with u c1u1ciuicmumoui00ui0 This leaves us with ciuioui0 Because uiui1 this means that ci0 Because this holds for any i it follows that the vectors in Rquotare linearly independent 0 Orthonormal vectors in Rquot form a basis in Rquot 0 Just rememberthe definition of a basis Orthogonal projection Consider a vector x in Rquot and a subspace V in Rquot Then we can write XX XJ where x is in V and XL is perpendicularto V and this representation is unique The vector x is called the orthogonal projection of x onto V denoted by proiju1oxuiumoxum Orthogonal complement Let V be a subspace of Rquot The orthogonal complement TxeR vox0 for all v in V Properties of the orthogonal complement Consider a subspace V of Rquot a The orthogonal complement T of V is a subspace of Rquot a If Txprojvx then TketT a subspace of Rquot b The intersection of V and W consists of the zero vector alone V VLO a If a vector x is in V and T then x is orthogonal to itself xox0 c dimVdimVJ39n a dimVlmT dimVJ39kerT so by the ranknullity theorem L dimVdimVJ39lmTkerTn V Pythagorean Theorem XY2X2Y21 iffX39FO Inequality of the magnitude of projvx Let V be a subspace of Rquot and xeRquot Then IIPFOJvXIISIIXII IIPFOJvXIIIIX1iffX V CauchySchwarz inequality If x and y are vectors in Rquot then xoysxy iff x amd y are parallel Angle between two vectors Consider two nonzero vector x and y in Rquot The angle earcosxoyxy Let xyeRquot The XYSXY XYISXIIYI X2X22XY39II2 S llxl22XIYIIIIYI2XIIIIYII2 X 2X YIIYI SX 2llxlllllll rllyll Section 52 The GramSchmidt process Consider a basis V1Vm of a subspace V of Rquot For i2m we resolve the vector vj into its components parallel and perpendicular to the span of the preceding vectors V1vJ11 U1V1V1 UJVJ39U139VJU13939Uj139Vjuj1 QR Factorization MQR where M is the old B Matrix and Q is the orthonormal basis R is the change of basis Matrix Section 53 Orthogonal transformation preserve orthogonality TVTWTVTW TVTWTVWVWVWTVTW Orthogonal transformations and orthonormal bases a TRquotgtRquot is orthogonal iff the vectors Te1Te form an orthonormal basis of Rquot b An nxn matrix A is orthogonal iff its columns form an orthonormal basis of Rquot Properties of Orthogonal matrices Consider an nxn matrix A Then the following statements are equivalent 1 A is an orthogonal matrix 2 The transformation LxAx preserves length that is Axx for all xeRquot 3 The column of A form an othonormal basis of Rquot 4 ATAIn 5 A391AT Section 55 Properties of Inner Product ltfggtltgfgt ltfhggtltfggtlthggt ltkfggtkltfggt ltffgtgt O for all nonzero f in V A linear space endowed with an inner product is called an inner product space General Inner Products 1 Cabcontinuoufs functions on interval abeR If fgeCab then ltfggtlabftgtdt a x n 2 Let ABeRquotX39 then ltABgttrATBtrBTA 3 L2xxiiiquotf such that Ziquotfi1xi2 converges ltxygtZiquotfi1xiyi Norm fsqrtltffgt Generalization f92llf292 GramSchmidt Process CauchySchwarz lltf9gtlf9 Orthogonal Projections Section 72 ml rt If A is an nxn matrix then detAAis a polynomial of degree n of the form AquottrAAquot391 detA This called the characteristic polynomial of A denoted by fAA We say that an eigenalue A0 of a square matrix A has a algebraic multiplicity k if A0 is a root of multiplicity k of the characteristic polynomial fAA meaning that we can write fAAAoAk9 for some polynomial gA with gA0 O Section 73 Consider an eigenvalue A of an nxn matrix A Then the kernel of the matrix AA is called the eigenspace associated with A denoted by E EkerAAv in Rquot AvAv Note that the eigenvectors with eigenvalue A are the nonzero vectors of eigenspaceE Consider an eigenvectr A of an nxn matrix A The dimension of eigenspaceEkerAA is call the geometric multiplicity of eigenvalue A Thus the geometric multiplicity is the nullity of matrix AAln or nrankAA Cosider an nxn matrix A A basis of Rquot consisting of eigenvectors of A is called an eigenbasis for A Suppose matrix A is similar to B Then a Matrices A and B have the same characteristic polynomial that is fAfB b RankArankB and the nullityAnullityB c Matrices A and B have the same eigenvalues with the same algebraic and geometric multiplicities However the eigenvectors need not be the same d Matrices A and B have the same determinant and trace detAdetb and trAtrB Section 74 An xn matrix A is called diagonalizable if A is similar to some matrix D that is there exist an 1 invertible nxn matrix such that 839 AS is diagonal A scalar A is called an eigenvalue of T if there exist a nonzero element f 6 V such that Tf Af Such an f is called an eigenvector of T quot rmly 15 r rm m W H 2 Suppose that V is finite dimensional Then a basis D of V consisting of eigenvectors of T is called an eigenbasis for T We say that transformation T is diagonalizable if the matrix of T with respect to some basis is diagonal To compute the powers At of a diagonalizable matrix A where t is a positive integer proceed as follows ASDS391 and A SD S391To compute raise the diagonal entries of D to the tth power Section 75 A complex nxn matrix has n complex eigenvalue if they are counted with their algebraic multiplicities 1 Hr MM r u Consider an nxn matrix A with complex eigenvalue A1 A2 TrA12 DetAA1A2An Problem that I need to work Test 1 Geometric Interpretations Test 2 BSquotAS Not SAS391 Test 3 Change of Basis Matrix Diagonalization and eigenbasis of L

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