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Lecture 1Subscript and summation notationThis materialis covered in Section 7 of the course notes.The problemHow do we write an arbitrary quadratic?What about a quartic?Two equations in two unknowns?Three equations in three unknowns?What if we run out of letters?A solution:subscriptsA better way to write a quartic might bea4x4+a3x3+· · ·+a1x+a0.In the same vein, a better way to write our three equations couldbeDouble-subscriptsTo be truly efficient in describing a system of 3 equations in 4unknowns, we’d like to use just three letters:one for the variables,one for the coefficients, and one for the right-hand sides.How?Double subscripts:Now it doesn’t really matter how many equations or unknowns:mequations innunknowns becomes
More efficient:a quantified variableWe can stillbe a bit more efficient by describing, generically, thejthequation and saying what valuesjis allowed to take:Sigma notationTo get rid of the sums, we use shorthand.The sigma notationnXi=mEimeansEm+Em+1+· · ·+Enand reads as “The sum asigoesfrommtonofEi.”5Xi=1aimeans4Xj=2j2means4Xm=1dmmmeansSigma notation in actionSo an efficient way to describe an arbitrary polynomialwould beOr a really efficient way to write downmequations innunknownswould beLecture 2Introducing matricesThis materialis covered in Workbook 7.1 of the course notes.
Recap (Lecture 1)To organise multiple unknowns in a systematic way, we usesubscripted variables, likea1,a2, . . .The sigma notationfXi=sEimeansEs+Es+1+· · ·+Ef1+Efand reads “the sum asigoes fromstofofEi.”Definition of matricesDefinitionAmatrixis a rectangular grid of numbers.If the grid hasmrowsandncolumns, then we callit anm×nmatrix (we say “mbynmatrix”).ExamplesNotation for matricesConvention:the matrix is calledA(capitalletter) and then thenumbers in the grid are calledai,j(lower-case letter).ExamplesA=101425112B=10π0.12.7ln(5)32ThenAis×andBis×.a2,3=a1,2=b1,3=b2,2=Terminology.We callthe numbers in a matrix itsentries.If we want to single out the entry in the 4throw and 7thcolumn,we callit the 4,7-entry.We callthe entry in theithrow andjthcolumn thei,j-entry.
Using matrices:systems of equationsConsider the system of equations7x13x2+ 5x3= 84x1+ 4x2+x3=1x1+x3= 5.We can encode allthe important information in this system in a3×4 matrix:Types of matrices:row and columnArow matrixorrow vectoris a matrix with just one row, so it’s1×n.Eg:Acolumn matrixorcolumn vectoris a matrix with just onecolumn, so it’sn×1.Eg:Types of matrices:squareAsquare matrixis one that has the same number of rows ascolumns.It isn×n.Eg:The number of rows (and columns)nis called thesizeororderofthe square matrix.Square matrices:Leading diagonaland traceTheleading diagonalin a square matrix is the entries from top leftto bottom right —.1123245011527Thetraceof a square matrix of ordernis the sum of the entries onthe leading diagonal.That istr(A) =.Youcannottake the trace of a matrix that isn’t square.
Triangular matricesA square matrix isupper triangularif allthe entries below theleading diagonalare zero.A square matrix islower triangularif allthe entries above theleading diagonalare zero.Diagonalmatrices and the zero matrixA square matrix isdiagonalif the only nonzero entries are on theleading diagonal.This is the same as saying it is bothand.For anymandnthem×nzero matrix0m,nis the matrix. . .. . .......We write 0minstead of 0m,m.Describe these matrices (give the trace if possible).A=132111430B=110011C=1203D=100000120E=100030001F=111111111(A)(B)(C)(D)(E)(F)Equality of matricesWe say that matricesAandBareequaland writeA=Bif:They are the same shape (bothm×n, say), andEach entry ofAis equalto the corresponding entry ofB.Examples:1331a33b1331a23b134312a33b
Adding matricesWe can add matricesAandBtogether provided they are the sameshape (bothm×nsay).Thei,j-entry ofA+Bis thei,j-entry ofAplus thei,j-entry ofB; that is,ai,j+bi,j.112322+132005=1811+3321=1811+334211=Subtracting matricesSubtracting matrices is just like adding them:they have to be thesame, size, and then we just do the subtraction separately in eachentry.112322132005=18113321=1811334211=The transpose of a matrixThetransposeof a matrixAis the matrixATwhose columns arethe rows ofA.Another way to write it isati,j=.Examples:134271222T=234567T=Symmetric matricesA matrixAis calledsymmetricif it is equalto its transpose.Thatis, ifA=AT.Another way to write this is thatai,j=for alli,j.Examples.20013202411162313791951234254736894790