University of Sheffield



Module title & Course codeMatrices and GeometryMAS 171LecturerK. OhktianiCourse descriptionThis unit will introduce the concept of a matrix, along with basic rules for arithmetic with matrices; it will also demonstrate the use of matrices for systematic manipulation of linear equations. It will introduce the important notions of determinants, eigenvalues and eigenvectors, and some particular classes of matrices with special properties. The unit will also introduce basic ideas of co-ordinate geometry, particularly Cartesian co-ordinates, emphasising links between geometry and algebra. It will cover formal mathematical representations of straight lines and of circles and other conics, and techniques for obtaining related geometrical objects such as tangents. University course contentA Level course contentModuleFurther module only?Solving simultaneous equations by Gaussian eliminationLinear dependence and IndependenceReduction to Row Echelon FormAlgebra of matricesTranspose of a matrixIdentity and inverse matricesGauss-Jordan eliminationDeterminants of matricesThe vector cross productApplication of determinants to linear independenceForming the adjoint matrixEigenvalues and EigenvectorsLinear independence of eigenvectorsSymmetric and Orthogonal matrices2d Co-ordinate Geometry – equations of straight linesIntersections of lines and perpendicular linesMidpoints and distancesCirclesTwo circle compound motionThe parabolaTangents to the parabolaThe reflection property of the parabolaParametric form of the parabolaThe ellipseThe hyperbolaQuadratic curvesAlthough Gaussian elimination itself does not feature on the A level specification, students of modules FP1 or FP3 will be familiar with the idea of turning a system of simultaneous equations into a single matrix equation. Students of module D2 are perhaps best prepared for this as the augmented matrix will come as little surprise to students who have seen a simplex tableau. The simplex method also involves using row operations in exactly the same way. Students of FP3 may have some general appreciation of the usefulness of triangular matrices as they may have calculated their determinants.These are completely new concepts to all students. Students of FP1/FP3 may have and advantage as they should know the abstract definition of a linear transformation so will be familiar with the word “linear” being applied to a linear combination of vectors. Few students, however, will have a deep or secure understanding of this area of the specification as it is rarely required. All students of C4, however, should have little trouble adding vectors and multiplying by a scalar.Students of module D2 will be used to reducing matrices to a different form via row operations. Row echelon form itself will be completely new to all studentsStudents of modules FP1 and FP3 should be able to add, scalar multiply and matrix multiply matrices of any size. They should know about the dimensions of a matrix. Subscript notation is not used to describe the entries of a matrix. Students are unlikely to know the words “associative” and “distributive” but may well know the word “non-commutative” and should at least know that matrix multiplication is not commutative. All students study module C4 and will understand vectors - but they are always column vectors as defined here (row vectors as defined here would only be used to denote the co-ordinates of a point at A level). As mentioned earlier, students of FP1/FP3 should be familiar with the matrix form of a system of simultaneous linear equations.Only students of module FP3 will know how to take the transpose of a matrix. They should also know the definition of “symmetric” but not “skew-symmetric”. Although they should know that (AB)T=BTAT, they will not know that ATA is always symmetric (proofs of this type are not beyond the scope of FP3, though).The identity matrix will be familiar to students of FP1 and FP3. Students of FP1 can form 2x2 inverse matrices. Students of FP3 know the word “diagonal” and can form 3x3 inverse matrices. Students of FP1 know the words “singular” and “non-singular” and how these are connected to the determinant.It would be rare but not impossible for a student of module FP3 to know this method. Again, students of module D2 have an advantage in that they can do the simplex algorithm. In the 3x3 case, students of module FP3 are likely to know how to form the matrix of cofactors and the adjoint matrix so the formula given should not be a surprise.Students of FP1 can calculate a 2x2 determinant. Students of FP3 can calculate minors, cofactors and 3x3 determinants. Neither students are likely to have considered these as denominators arising from solving simultaneous equations. Students of FP3 are extremely unlikely to know the effect of row operations on the determinant. However, it is likely that students of FP3 will know how easy it is to calculate the determinants of upper/lower triangular and diagonal matrices.Students of module FP3 should be able to calculate the cross product and should know the formula given in terms of 3x3 detereminantsThis will be new to all studentsAll students of module FP3 know this method for 3x3 inverse matricesStudents of FP3 know the Eigenvalue and Eigenvector equation. They may use the words “characteristic polynomial” to refer to the Eigenvalue equation. They will not know the trace of a matrixThis is new to all students. However, students of module FP3 do know the related fact that the eigenvectors of symmetric matrices are orthogonal. They will not have considered eigenvectors of iterated matrices.Students of FP3 should know that the eigenvectors of a symmetric matrix are orthogonal. They will not have considered the possibility of non-real eigenvectors. They should know the meaning of “orthogonal” and “orthonormal” when applied to vectors. They should know that an “orthogonal” matrix has orthonormal columns (and as such be clear on the different usage of the word “orthogonal for matrices and vectors). They should also know that the transpose of an orthogonal matrix is the same as its inverse.All students should learn ax+by+c=0, y=mx+c and y-y1=m(x-x1) in module C1. It is quite likely that they will also know y-y1y2-y1=x-x1x2-x1 although this is not compulsory. Nonetheless, this should be straightforward to any maths student. They should know the gradient formula but only students offering STEP or AEA are likely to have connected this with tanθ.All students should have little trouble with intersections and the condition that the gradient of a perpendicular line is the negative reciprocal of the original gradient is known to all students from module C1 and is used often throughout A level in calculating the equations of normal to curves. All students study module C3 and consequently should have little to fear from the use of cotθ.The formulae for midpoint and distance between two points should be known to all students from modules C2. However the formula for distance from a point to a line will likely be new to all students.The circle theorem about tangents should be known to all GCSE students. The equation of a circle, intersection of line and circles and the use of completing the square to find the centre and radius of a circle should be routine for all C2 students. They are unlikely to have examined the intersection of two circles. The parametric form will come as little surprise to students of FP3 who know the parametric equations of the ellipse. All students study parametric equations on module C4.This will be new to all students. In general, unless a student has studied AEA or STEP, they are likely to have poor geometric intuition so this may well prove difficult for themThe form y2=4ax and the focus directrix property should be known to all FP1 students. It is rare that the parabola will be moved away from the origin or found in a different orientation, though.This style of question will be quite unfamiliar. At A level, students are normally given a point on the curve at which to calculate a tangent.It would not be uncommon for a student of FP1 to have encountered this but it is not compulsoryThis will be known to students of FP1 and should be of little difficulty to all students from C4 parametric equationsStudents of FP3 should know the Cartesian and parametric equations of the ellipse, the focus-directrix property and should have little trouble finding tangents. It is not uncommon for them to know that the sum of the distances of a point on the ellipse to the two foci is constant (however, students do find locus problems very difficult). They are very unlikely to know the reflection property.Students of FP3 should know the Cartesian and parametric equations of the hyperbola, the focus-directrix property and should have little trouble finding tangents. They should also know the equations of the asymptotes. They are very unlikely to know the reflection property or to have calculated tangents for a point not on the hyperbola. Students of FP1 study the rectangular hyperbola – they will know xy=c2 and the corresponding parametric equations but will not have considered the significance of the asymptotes being at right anglesIt is unlikely that any student will have considered the equations of ellipses and hyperbolae not in their standard position and orientationFP1/FP3/D2FP1/FP3/C4D2FP1/FP3/C4FP3FP1/FP3FP3/D2FP1/FP3FP3N/AFP3FP3N/AFP3C1C1/C3C2C2/FP3/C4N/AFP1N/AN/AFP1/C4FP3FP3/FP1N/AYYYYYYYYYN/AYYN/AYNNNNN/AYN/AN/AYYYN/A ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download