Algebra And Geometry
Jump to year
Paper 1, Section I,
comment(i) The spherical polar unit basis vectors and in are given in terms of the Cartesian unit basis vectors and by
Express and in terms of and .
(ii) Use suffix notation to prove the following identity for the vectors , and in :
Paper 1, Section I, B
commentFor the equations
find the values of and for which
(i) there is a unique solution;
(ii) there are infinitely many solutions;
(iii) there is no solution.
Paper 1, Section II, A
comment(i) Show that any line in the complex plane can be represented in the form
where and .
(ii) If and are two complex numbers for which
show that either or is real.
(iii) Show that any Möbius transformation
that maps the real axis into the unit circle can be expressed in the form
where and .
Paper 1, Section II, C
commentLet and be non-zero vectors in a real vector space with scalar product denoted by . Prove that , and prove also that if and only if for some scalar .
(i) By considering suitable vectors in , or otherwise, prove that the inequality holds for any real numbers and .
(ii) By considering suitable vectors in , or otherwise, show that only one choice of real numbers satisfies , and find these numbers.
Paper 1, Section II, C
commentLet be the linear map defined by
where and are positive scalar constants, and is a unit vector.
(i) By considering the effect of on and on a vector orthogonal to , describe geometrically the action of .
(ii) Express the map as a matrix using suffix notation. Find and in the case
(iii) Find, in the general case, the inverse map (i.e. express in terms of in vector form).
Paper 1, Section II, C
comment(i) Describe geometrically the following surfaces in three-dimensional space:
(a) , where
(b) , where .
Here and are fixed scalars and is a fixed unit vector. You should identify the meaning of and for these surfaces.
(ii) The plane , where is a fixed unit vector, and the sphere with centre and radius intersect in a circle with centre and radius .
(a) Show that , where you should give in terms of and .
(b) Find in terms of and .
Paper 3, Section I, D
commentWhat does it mean to say that groups and are isomorphic?
Prove that no two of and are isomorphic. [Here denotes the cyclic group of order .]
Give, with justification, a group of order 8 that is not isomorphic to any of those three groups.
Paper 3, Section I, D
commentProve that every permutation of may be expressed as a product of disjoint cycles.
Let and let . Write as a product of disjoint cycles. What is the order of
Paper 3, Section II,
commentWhat does it mean to say that a subgroup of a group is normal? Give, with justification, an example of a subgroup of a group that is normal, and also an example of a subgroup of a group that is not normal.
If is a normal subgroup of , explain carefully how to make the set of (left) cosets of into a group.
Let be a normal subgroup of a finite group . Which of the following are always true, and which can be false? Give proofs or counterexamples as appropriate.
(i) If is cyclic then and are cyclic.
(ii) If and are cyclic then is cyclic.
(iii) If is abelian then and are abelian.
(iv) If and are abelian then is abelian.
Paper 3, Section II, D
commentLet be an element of a finite group . What is meant by the order of ? Prove that the order of must divide the order of . [No version of Lagrange's theorem or the Orbit-Stabilizer theorem may be used without proof.]
If is a group of order , and is a divisor of with , is it always true that must contain an element of order ? Justify your answer.
Prove that if and are coprime then the group is cyclic.
If and are not coprime, can it happen that is cyclic?
[Here denotes the cyclic group of order .]
Paper 3, Section II, D
commentIn the group of Möbius maps, what is the order of the Möbius map ? What is the order of the Möbius map ?
Prove that every Möbius map is conjugate either to a map of the form (some ) or to the . Is conjugate to a map of the form
Let be a Möbius map of order , for some positive integer . Under the action on of the group generated by , what are the various sizes of the orbits? Justify your answer.
Paper 3, Section II, D
commentLet be a real symmetric matrix. Prove that every eigenvalue of is real, and that eigenvectors corresponding to distinct eigenvalues are orthogonal. Indicate clearly where in your argument you have used the fact that is real.
What does it mean to say that a real matrix is orthogonal ? Show that if is orthogonal and is as above then is symmetric. If is any real invertible matrix, must be symmetric? Justify your answer.
Give, with justification, real matrices with the following properties:
(i) has no real eigenvalues;
(ii) is not diagonalisable over ;
(iii) is diagonalisable over , but not over ;
(iv) is diagonalisable over , but does not have an orthonormal basis of eigenvectors.
1.I.1B
commentConsider the cone in defined by
Find a unit normal to at the point such that .
Show that if satisfies
and then
1.I.2A
commentExpress the unit vector of spherical polar coordinates in terms of the orthonormal Cartesian basis vectors .
Express the equation for the paraboloid in (i) cylindrical polar coordinates and (ii) spherical polar coordinates .
In spherical polar coordinates, a surface is defined by , where is a real non-zero constant. Find the corresponding equation for this surface in Cartesian coordinates and sketch the surfaces in the two cases and .
1.II.5C
commentProve the Cauchy-Schwarz inequality,
for two vectors . Under what condition does equality hold?
Consider a pyramid in with vertices at the origin and at , where , and so on. The "base" of the pyramid is the dimensional object specified by for .
Find the point in equidistant from each vertex of and find the length of is the centroid of .)
Show, using the Cauchy-Schwarz inequality, that this is the closest point in to the origin .
Calculate the angle between and any edge of the pyramid connected to . What happens to this angle and to the length of as tends to infinity?
1.II.6C
commentGiven a vector , write down the vector obtained by rotating through an angle .
Given a unit vector , any vector may be written as where is parallel to and is perpendicular to . Write down explicit formulae for and , in terms of and . Hence, or otherwise, show that the linear map
describes a rotation about through an angle , in the positive sense defined by the right hand rule.
Write equation in matrix form, . Show that the trace .
Given the rotation matrix
where , find the two pairs , with , giving rise to . Explain why both represent the same rotation.
1.II.7B
comment(i) Let be unit vectors in . Write the transformation on vectors
in matrix form as for a matrix . Find the eigenvalues in the two cases (a) when , and (b) when are parallel.
(ii) Let be the set of complex hermitian matrices with trace zero. Show that if there is a unique vector such that
Show that if is a unitary matrix, the transformation
maps to , and that if , then where means ordinary Euclidean length. [Hint: Consider determinants.]
1.II.8A
comment(i) State de Moivre's theorem. Use it to express as a polynomial in .
(ii) Find the two fixed points of the Möbius transformation
that is, find the two values of for which .
Given that and , show that a general Möbius transformation
has two fixed points given by
where are the square roots of .
Show that such a transformation can be expressed in the form
where is a constant that you should determine.
3.I.1D
commentGive an example of a real matrix with eigenvalues . Prove or give a counterexample to the following statements:
(i) any such is diagonalisable over ;
(ii) any such is orthogonal;
(iii) any such is diagonalisable over .
3.I.2D
commentShow that if and are subgroups of a group , then is also a subgroup of . Show also that if and have orders and respectively, where and are coprime, then contains only the identity element of . [You may use Lagrange's theorem provided it is clearly stated.]
3.II.5D
commentLet be a group and let be a non-empty subset of . Show that
is a subgroup of .
Show that given by
defines an action of on itself.
Suppose is finite, let be the orbits of the action and let for . Using the Orbit-Stabilizer Theorem, or otherwise, show that
where the sum runs over all values of such that .
Let be a finite group of order , where is a prime and is a positive integer. Show that contains more than one element.
3.II.6D
commentLet be a homomorphism between two groups and . Show that the image of , is a subgroup of ; show also that the kernel of , is a normal subgroup of .
Show that is isomorphic to .
Let be the group of real orthogonal matrices and let be the set of orthogonal matrices with determinant 1 . Show that is a normal subgroup of and that is isomorphic to the cyclic group of order
Give an example of a homomorphism from to with kernel of order
3.II.7D
commentLet be the group of real matrices with determinant 1 and let be a homomorphism. On consider the product
Show that with this product is a group.
Find the homomorphism or homomorphisms for which is a commutative group.
Show that the homomorphisms for which the elements of the form with , commute with every element of are precisely those such that
with an arbitrary homomorphism.
3.II.8D
commentShow that every Möbius transformation can be expressed as a composition of maps of the forms: and , where .
Show that if and are two triples of distinct points in , there exists a unique Möbius transformation that takes to .
Let be the group of those Möbius transformations which map the set to itself. Find all the elements of . To which standard group is isomorphic?
1.I.1C
commentConvert the following expressions from suffix notation (assuming the summation convention in three dimensions) into standard notation using vectors and/or matrices, where possible, identifying the one expression that is incorrectly formed:
(i) ,
(ii) ,
(iii) ,
(iv) ,
(v) .
Write the vector triple product in suffix notation and derive an equivalent expression that utilises scalar products. Express the result both in suffix notation and in standard vector notation. Hence or otherwise determine when and are orthogonal and .
1.I.2B
commentLet be a unit vector. Consider the operation
Write this in matrix form, i.e., find a matrix such that for all , and compute the eigenvalues of . In the case when , compute and its eigenvalues and eigenvectors.
1.II.5C
commentGive the real and imaginary parts of each of the following functions of , with real, (i) , (ii) , (iii) , (iv) , (v) ,
where is the complex conjugate of .
An ant lives in the complex region given by . Food is found at such that
Drink is found at such that
Identify the places within where the ant will find the food or drink.
1.II.6B
commentLet be a real matrix. Define the rank of . Describe the space of solutions of the equation
organizing your discussion with reference to the rank of .
Write down the equation of the tangent plane at on the sphere and the equation of a general line in passing through the origin .
Express the problem of finding points on the intersection of the tangent plane and the line in the form . Find, and give geometrical interpretations of, the solutions.
1.II.7A
commentConsider two vectors and in . Show that a may be written as the sum of two vectors: one parallel (or anti-parallel) to and the other perpendicular to . By setting the former equal to , where is a unit vector along , show that
Explain why this is a sensible definition of the angle between and .
Consider the vertices of a cube of side 2 in , centered on the origin. Each vertex is joined by a straight line through the origin to another vertex: the lines are the diagonals of the cube. Show that no two diagonals can be perpendicular if is odd.
For , what is the greatest number of mutually perpendicular diagonals? List all the possible angles between the diagonals.
1.II.8A
commentGiven a non-zero vector , any symmetric matrix can be expressed as
for some numbers and , some vector and a symmetric matrix , where
and the summation convention is implicit.
Show that the above statement is true by finding and explicitly in terms of and , or otherwise. Explain why and together provide a space of the correct dimension to parameterise an arbitrary symmetric matrix .
3.I.1D
commentLet be a real symmetric matrix with eigenvalues . Consider the surface in given by
Find the minimum distance between the origin and . How many points on realize this minimum distance? Justify your answer.
3.I.2D
commentDefine what it means for a group to be cyclic. If is a prime number, show that a finite group of order must be cyclic. Find all homomorphisms , where denotes the cyclic group of order . [You may use Lagrange's theorem.]
3.II.5D
commentDefine the notion of an action of a group on a set . Assuming that is finite, state and prove the Orbit-Stabilizer Theorem.
Let be a finite group and the set of its subgroups. Show that defines an action of on . If is a subgroup of , show that the orbit of has at most elements.
Suppose is a subgroup of and . Show that there is an element of which does not belong to any subgroup of the form for .
3.II.6D
commentLet be the group of Möbius transformations of and let be the group of all complex matrices with determinant 1 .
Show that the map given by
is a surjective homomorphism. Find its kernel.
Show that every not equal to the identity is conjugate to a Möbius map where either with , or . [You may use results about matrices in , provided they are clearly stated.]
Show that if , then is the identity, or has one, or two, fixed points. Also show that if has only one fixed point then as for any
3.II.7D
commentLet be a group and let for all . Show that is a normal subgroup of
Let be the set of all real matrices of the form
with . Show that is a subgroup of the group of invertible real matrices under multiplication.
Find and show that is isomorphic to with vector addition.
3.II.8D
commentLet be a real matrix such that , and , where is the transpose of and is the identity.
Show that the set of vectors for which forms a 1-dimensional subspace.
Consider the plane through the origin which is orthogonal to . Show that maps to itself and induces a rotation of by angle , where . Show that is a reflection in if and only if has trace 1 . [You may use the fact that for any invertible matrix B.]
Prove that .
1.I.1B
commentThe linear map represents reflection in the plane through the origin with normal , where , and referred to the standard basis. The map is given by , where is a matrix.
Show that
Let and be unit vectors such that is an orthonormal set. Show that
and find the matrix which gives the mapping relative to the basis .
1.I.2C
commentShow that
for any real numbers . Using this inequality, show that if and are vectors of unit length in then .
1.II.5B
commentThe vector satisfies the equation
where is a matrix and is a column vector. State the conditions under which this equation has (a) a unique solution, (b) an infinity of solutions, (c) no solution for .
Find all possible solutions for the unknowns and which satisfy the following equations:
in the cases (a) , and (b) .
1.II.6A
commentExpress the product in suffix notation and thence prove that the result is zero.
Silver Beard the space pirate believed people relied so much on space-age navigation techniques that he could safely write down the location of his treasure using the ancient art of vector algebra. Spikey the space jockey thought he could follow the instructions, by moving by the sequence of vectors one stage at a time. The vectors (expressed in 1000 parsec units) were defined as follows:
Start at the centre of the galaxy, which has coordinates .
Vector a has length , is normal to the plane and is directed into the positive quadrant.
Vector is given by , where .
Vector has length , is normal to and , and moves you closer to the axis.
Vector .
Vector has length . Spikey was initially a little confused with this one, but then realised the orientation of the vector did not matter.
Vector has unknown length but is parallel to and takes you to the treasure located somewhere on the plane .
Determine the location of the way-points Spikey will use and thence the location of the treasure.
1.II.7A
commentSimplify the fraction
where is the complex conjugate of . Determine the geometric form that satisfies
Find solutions to
and
where is a complex variable. Sketch these solutions in the complex plane and describe the region they enclose. Derive complex equations for the circumscribed and inscribed circles for the region. [The circumscribed circle is the circle that passes through the vertices of the region and the inscribed circle is the largest circle that fits within the region.]
1.II.8C
comment(i) The vectors in satisfy . Are necessarily linearly independent? Justify your answer by a proof or a counterexample.
(ii) The vectors in have the property that every subset comprising of the vectors is linearly independent. Are necessarily linearly independent? Justify your answer by a proof or a counterexample.
(iii) For each value of in the range , give a construction of a linearly independent set of vectors in satisfying
where is the Kronecker delta.
3.I.1D
commentState Lagrange's Theorem.
Show that there are precisely two non-isomorphic groups of order 10 . [You may assume that a group whose elements are all of order 1 or 2 has order .]
3.I.2D
commentDefine the Möbius group, and describe how it acts on .
Show that the subgroup of the Möbius group consisting of transformations which fix 0 and is isomorphic to .
Now show that the subgroup of the Möbius group consisting of transformations which fix 0 and 1 is also isomorphic to .
3.II.5D
commentLet be the dihedral group of order 12 .
i) List all the subgroups of of order 2 . Which of them are normal?
ii) Now list all the remaining proper subgroups of . [There are of them.]
iii) For each proper normal subgroup of , describe the quotient group .
iv) Show that is not isomorphic to the alternating group .
3.II.6D
commentState the conditions on a matrix that ensure it represents a rotation of with respect to the standard basis.
Check that the matrix
represents a rotation. Find its axis of rotation .
Let be the plane perpendicular to the axis . The matrix induces a rotation of by an angle . Find .
3.II.7D
commentLet be a real symmetric matrix. Show that all the eigenvalues of are real, and that the eigenvectors corresponding to distinct eigenvalues are orthogonal to each other.
Find the eigenvalues and eigenvectors of
Give an example of a non-zero complex symmetric matrix whose only eigenvalues are zero. Is it diagonalisable?
3.II.8D
commentCompute the characteristic polynomial of
Find the eigenvalues and eigenvectors of for all values of .
For which values of is diagonalisable?
1.I.1B
comment(a) Write the permutation
as a product of disjoint cycles. Determine its order. Compute its sign.
(b) Elements and of a group are conjugate if there exists a such that
Show that if permutations and are conjugate, then they have the same sign and the same order. Is the converse true? (Justify your answer with a proof or counterexample.)
1.I.2D
commentFind the characteristic equation, the eigenvectors , and the corresponding eigenvalues of the matrix
Show that spans the complex vector space .
Consider the four subspaces of defined parametrically by
Show that multiplication by maps three of these subspaces onto themselves, and the remaining subspace into a smaller subspace to be specified.
1.II.5B
comment(a) In the standard basis of , write down the matrix for a rotation through an angle about the origin.
(b) Let be a real matrix such that and , where is the transpose of .
(i) Suppose that has an eigenvector with eigenvalue 1 . Show that is a rotation through an angle about the line through the origin in the direction of , where trace .
(ii) Show that must have an eigenvector with eigenvalue 1 .
1.II.6A
commentLet be a linear map
Define the kernel and image of .
Let . Show that the equation has a solution if and only if
Let have the matrix
with respect to the standard basis, where and is a real variable. Find and for . Hence, or by evaluating the determinant, show that if and then the equation has a unique solution for all values of .
1.II.7B
comment(i) State the orbit-stabilizer theorem for a group acting on a set .
(ii) Denote the group of all symmetries of the cube by . Using the orbit-stabilizer theorem, show that has 48 elements.
Does have any non-trivial normal subgroups?
Let denote the line between two diagonally opposite vertices of the cube, and let
be the subgroup of symmetries that preserve the line. Show that is isomorphic to the direct product , where is the symmetric group on 3 letters and is the cyclic group of order 2 .
1.II.8D
commentLet and be non-zero vectors in . What is meant by saying that and are linearly independent? What is the dimension of the subspace of spanned by and if they are (1) linearly independent, (2) linearly dependent?
Define the scalar product for . Define the corresponding norm of . State and prove the Cauchy-Schwarz inequality, and deduce the triangle inequality.
By means of a sketch, give a geometric interpretation of the scalar product in the case , relating the value of to the angle between the directions of and .
What is a unit vector? Let be unit vectors in . Let
Show that
(i) for any fixed, linearly independent and , the minimum of over is attained when for some ;
(ii) in all cases;
(iii) and in the case where .
3.I.1A
commentThe mapping of into itself is a reflection in the plane . Find the matrix of with respect to any basis of your choice, which should be specified.
The mapping of into itself is a rotation about the line through , followed by a dilatation by a factor of 2 . Find the matrix of with respect to a choice of basis that should again be specified.
Show explicitly that
and explain why this must hold, irrespective of your choices of bases.
3.I.2B
commentShow that if a group contains a normal subgroup of order 3, and a normal subgroup of order 5 , then contains an element of order 15 .
Give an example of a group of order 10 with no element of order
3.II.5E
comment(a) Show, using vector methods, that the distances from the centroid of a tetrahedron to the centres of opposite pairs of edges are equal. If the three distances are and if are the distances from the centroid to the vertices, show that
[The centroid of points in with position vectors is the point with position vector
(b) Show that
with , is the equation of a right circular double cone whose vertex has position vector a, axis of symmetry and opening angle .
Two such double cones, with vertices and , have parallel axes and the same opening angle. Show that if , then the intersection of the cones lies in a plane with unit normal
3.II.6E
commentDerive an expression for the triple scalar product in terms of the determinant of the matrix whose rows are given by the components of the three vectors .
Use the geometrical interpretation of the cross product to show that , will be a not necessarily orthogonal basis for as long as .
The rows of another matrix are given by the components of three other vectors . By considering the matrix , where denotes the transpose, show that there is a unique choice of such that is also a basis and
Show that the new basis is given by
Show that if either or is an orthonormal basis then is a rotation matrix.
3.II.7B
commentLet be the group of Möbius transformations of and let be a set of three distinct points in .
(i) Show that there exists a sending to to 1 , and to .
(ii) Hence show that if , then is isomorphic to , the symmetric group on 3 letters.
3.II.8B
comment(a) Determine the characteristic polynomial and the eigenvectors of the matrix
Is it diagonalizable?
(b) Show that an matrix with characteristic polynomial is diagonalizable if and only if .
1.I.1B
comment(a) State the Orbit-Stabilizer Theorem for a finite group acting on a set .
(b) Suppose that is the group of rotational symmetries of a cube . Two regular tetrahedra and are inscribed in , each using half the vertices of . What is the order of the stabilizer in of ?
1.I.2D
commentState the Fundamental Theorem of Algebra. Define the characteristic equation for an arbitrary matrix whose entries are complex numbers. Explain why the matrix must have three eigenvalues, not necessarily distinct.
Find the characteristic equation of the matrix
and hence find the three eigenvalues of . Find a set of linearly independent eigenvectors, specifying which eigenvector belongs to which eigenvalue.
1.II.5B
comment(a) Find a subset of the Euclidean plane that is not fixed by any isometry (rigid motion) except the identity.
Let be a subgroup of the group of isometries of a subset of not fixed by any isometry except the identity, and let denote the union . Does the group of isometries of contain ? Justify your answer.
(b) Find an example of such a and with .
1.II.6B
comment(a) Suppose that is a Möbius transformation, acting on the extended complex plane. What are the possible numbers of fixed points that can have? Justify your answer.
(b) Show that the operation of complex conjugation, defined by , is not a Möbius transformation.
1.II.7B
comment(a) Find, with justification, the matrix, with respect to the standard basis of , of the rotation through an angle about the origin.
(b) Find the matrix, with respect to the standard basis of , of the rotation through an angle about the axis containing the point and the origin. You may express your answer in the form of a product of matrices.
1.II.8D
Define what is meant by a vector space over the real numbers . Define subspace, proper subspace, spanning set, basis, and dimension.
Define the sum and intersection of two subspaces and of a vector space . Why is the intersection never empty?
Let and let