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More on Inverse Last Week Review • Matrix – – – – – Rule of addition Rule of multiplication Transpose Main Diagonal Dot Product • Block Multiplication • Matrix and Linear Equations – Basic Solution • X1 + X0 – Linear Combination – All solutions of LES • Inverse – Det – Matrix Inversion Method • Double matrix Warm Up • Find the inverse of • Using matrix inversion method – [ A I ] [ I A-1 ] Solution • Start with the double matrix • Swap 1 with 2 • R2 – 2R1, R3 – R1 • More to Reduced Row Echelon Form PROPERTIES OF INVERSE Transpose and Inverse • If A is invertible, show that – AT is also invertible – (AT)-1 = (A-1)T Solution • A-1 exists – Its transpose is the inverse of AT • So AT(A-1)T = (A-1A)T = IT = I (A-1)TAT = (AA-1)T = IT = I Inverse of Multiplication • If A and B are invertible, show that – AB is also invertible – (AB)-1 = B-1A-1 Solution • Assume that (AB)-1 exists – And it is B-1A-1 • (B-1A-1)(AB) = B-1(A-1A)B = B-1IB = B-1B = I • (AB)(B-1A-1) = A(BB-1)A-1 = AIA-1 = AA-1 = I • Hence, it is actually the inverse Rule of Inverse Inverse Equivalence • A is invertible • The homogeneous system AX = 0 has only the trivial solution X = 0 • A can be carried to the identity matrix In by elementary row operation • The system AX=B has at least one solution X • There exists an n x n matrix C such that AC = In ELEMENTARY MATRICES Elementary Matrix • A matrix that can be obtained from I by single elementary row operation • Example Elementary Operation • Interchange two equations • Multiply one equation with a number • Add a multiple of one equation to a equation Lemma • If an elementary row operation is performed on an matrix • The result is where is the elementary matrix is obtained by performing the same operation on identity matrix. Inverse of elementary operation • Each operation has an inverse – Also an elementary operation • So are the elementary matrix Operation Inverse Interchange row p and q Interchange row q and p Multiply row p by k != 0 Multiply row p by 1/k Add k times row p to row q != p Subtract k times row p to row q Inverse of Elementary Matrix • Hence, each elementary matrix E has its inverse • The inverse change E back to I Lemma 2 • Every elementary matrix E is invertible – Its inverse is also an elementary matrix • Of the same type as well • It also corresponds to the inverse of the row operation that produce E Inverse and Rank • Suppose that A B by a series of elementary row operation • Hence – A E1A E2E1A EkEk-1…E2E1A B • i.e., A UA = B – Where U = EkEk-1…E2E1 • U is invertible – Why? Finding U • AB by some elementary row operations • Perform the same operations on I • Doing the same thing just like the matrix inversion algorithm • [A I] [B U] Theorem: Property of U • Suppose that A is m x n and A B by some sequence of elementary row operations – B = UA where U is m x m invertible matrix – U can be computed by [A I] [B U] using the same operations – U = EkEk-1…E2E1 where each Ei is the elementary matrix corresponding to the elementary row operation U and -1 A • Suppose that A is invertible – We know that A I – So, let B be I – Hence, [A I] [I U] • I = UA • i.e., U = A-1 • This is exactly the matrix inversion algorithm – But, A-1 =U = EkEk-1…E2E1 – Hence A = (A-1)-1 = (EkEk-1…E2E1)-1 = E1-1E2-1…Ek-1-1Ek-1 • This means that every invertible matrix is a product of elementary matrices!!! Theorem 2 • A square matrix is invertible if and only if it is a product of elementary matrices. TRANSFORMATION Ordered n-tuple (Vector) • Let be the set of real number • If n >= 1, an ordered sequence – (a1,a2,..,an) is called an ordered n-tuple denotes the set of all ordered n-tuples • The ordered n-tuple is also called vectors Transformation • A function T from • Written T: to domain codomain • To describe T, we must give the definition of all T(X) for every X in • T and S is the same if T(X) = S(X) for every X – That is the definition of T Matrix Transformation • A transformation such that – T(X) is AX • Called the matrix transformation induced by A – If A = 0, it is called the zero transformation – If A = I, it is called the identify transformation Example • X-expansion • Induced by Example • Reflection • Induced by Example • X-shear • Induced by Translation is not Linear Transform • Translation – T(X) = X + w • If it is, then – X + w = AX for some A – What if a = 0? Linear Transformation • A transformation is called a linear transformation when – T(X + Y) = T(X) + T(Y) – T(aX) = aT(X) Linear Transform and Matrix Transform • Let T: RN RM be a transformation – T is linear if and only if it is a matrix transformation – If T is linear, then T is induced by a unique matrix A Composition • Transform of a transform • ST = S(T(X)) Composition • If R,S,T are linear transformation – Compositions of them are also linear – Is associative • (since it is matrix transform) Inverse through transform • Inverse of the transform is the inverse of the function • Hence, domain and codomain must be the same • Given a linear transformation – It’s inverse is induced by A-1