Transcription of Chapters 7-8: Linear Algebra - University of Arizona
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Chapters 7-8: Linear Algebra - University of Arizona
Linear systems of equationsInverse of a matrixEigenvalues and eigenvectorsChapters 7-8: Linear AlgebraSections , & 7-8: Linear AlgebraLinear systems of equationsInverse of a matrixEigenvalues and eigenvectorsDefinitionsSolutions1. Linear systems of equationsAlinear systemof equations of the forma11x1+a12x2+ +a1nxn=b1a21x1+a22x2+ +a2nxn=b2 am1x1+am2x2+ +amnxn=bmcan be written in matrix form asAX=B,whereA= a11a12 a1na21a22 amn ,X= ,B= Chapters 7-8: Linear AlgebraLinear systems of equationsInverse of a matrixEigenvalues and eigenvectorsDefinitionsSolutionsSolution (s) of a Linear system of equationsGiven a matrixAand a vectorB,asolutionof the systemAX=Bis a vectorXwhich satisfies the equationAX= not in the column space ofA, then the systemAX=Bhasno solution. One says that the system isnotconsistent. In the statements below,we assume that thesystemAX=Bis the null space ofAis non-trivial, then the systemAX=Bhasmore than one systemAX=Bhas aunique solutionprovideddim(N(A)) = , by the rank theorem, rank(A)+dim(N(A)) =n(recallthatnis the number of columns ofA), the systemAX=Bhas aunique solutionif and only if rank(A)= 7-8: Linear AlgebraLinear systems of equationsInverse of a matrixEigenvalues and eigenvectorsDefinitionsSolutionsSolution (s) of a Linear system of equations (continued)A Linear system of the formAX= 0 is said to ofAX=0arevectors in the null space we know
i=1 aijCij = n j=1 aijCij where the cofactor Cij is given by Cij =(−1) i+j M ij, and the minor Mij is the determinant of the matrix obtained from A by “deleting” the i-th row and j-th column of A. Example: Calculate the determinant of A = ⎡ ⎣ 123 456 789 ⎤ ⎦. Chapters 7-8: Linear Algebra
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