Arbitrary Quadratic - Assignment
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Added on 2020-11-09
Arbitrary Quadratic - Assignment
Added on 2020-11-09
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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 in n unknowns becomes
More efficient:a quantified variableWe can stillbe a bit more efficient by describing, generically, thejthequation and saying what values jis allowed to take:Sigma notationTo get rid of the sums, we use shorthand.The sigma notationnXi=mEimeans Em+ Em+1+ · · · + Enand reads as “The sum as igoesfrom m to n of Ei.”5Xi=1aimeans4Xj=−2j2means4Xm=1dmmmeansSigma notation in actionSo an efficient way to describe an arbitrary polynomialwould beOr a really efficient way to write down m equations in n unknownswould 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, like a1, a2, . . .◮The sigma notationfXi =sEimeans Es+ Es+1+ · · · + Ef −1+ Efand reads “the sum as igoes from s to fof Ei.”Definition of matricesDefinitionA matrix is a rectangular grid of numbers.If the grid has m rowsand n columns, then we callit an m × n matrix (we say “m by nmatrix”).ExamplesNotation for matricesConvention:the matrix is called A (capitalletter) and then thenumbers in the grid are called ai ,j(lower-case letter).ExamplesA =10−142−51−12B =10π0.12.7ln(5)32Then A is×and B is×.◮a2,3=◮a1,2=◮b1,3=◮b2,2=Terminology.We callthe numbers in a matrix its entries.If we want to single out the entry in the 4throw and 7thcolumn,we callit the 4, 7-entry.We callthe entry in the ithrow and jthcolumn the i , j -entry.
Using matrices:systems of equationsConsider the system of equations7x1−3x2+ 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 columnA row matrix or row vector is a matrix with just one row, so it’s1 × n.Eg:A column matrix or column vector is a matrix with just onecolumn, so it’s n × 1.Eg:Types of matrices:squareA square matrix is one that has the same number of rows ascolumns.It is n × n.Eg:The number of rows (and columns) n is called the size or order ofthe square matrix.Square matrices:Leading diagonaland traceThe leading diagonalin a square matrix is the entries from top left—— to bottom right —.1−123245011527The trace of a square matrix of order n is the sum of the entries onthe leading diagonal.That istr(A) =.You cannot take the trace of a matrix that isn’t square.
Triangular matricesA square matrix is upper triangular if allthe entries below theleading diagonalare zero.A square matrix is lower triangular if allthe entries above theleading diagonalare zero.Diagonalmatrices and the zero matrixA square matrix is diagonalif the only nonzero entries are on theleading diagonal.This is the same as saying it is bothand.For any m and n the m × n zero matrix 0m,nis the matrix. . .. . .......We write 0minstead of 0m,m.Describe these matrices (give the trace if possible).A =13211−1430B =1100−11C =1203D =100000120E =10003000−1F =111111111(A)(B)(C)(D)(E)(F)Equality of matricesWe say that matrices A and B are equaland write A = B if:◮They are the same shape (both m × n, say), and◮Each entry of A is equalto the corresponding entry of B.Examples:133−1a33b133−1a23b1343−12a33b
Adding matricesWe can add matrices A and B together provided they are the sameshape (both m × n say).The i , j -entry of A + B is the i , j -entry of A plus the i , j -entry ofB ; that is, ai ,j+ bi ,j.1−12322+−132005=18−1−1+332−1=18−1−1+3342−11=Subtracting matricesSubtracting matrices is just like adding them:they have to be thesame, size, and then we just do the subtraction separately in eachentry.1−12322−−132005=18−1−1−332−1=18−1−1−3342−11=The transpose of a matrixThe transpose of a matrix A is the matrix ATwhose columns arethe rows of A.Another way to write it isati ,j=.Examples:134271222T=234567T=Symmetric matricesA matrix A is called symmetric if it is equalto its transpose.Thatis, if A = AT.Another way to write this is thatai ,j=for alli , j .Examples.200132024111623−1379−1951234254736894790
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