Linear Algebra for Engineering (2):
The Final Exam
There will be a 2-hours long final exam. There will be some choice;
you'll have to solve 4 out of 5 questions, or 5 out of 6, or something
like that. The questions will be multi-part, with the all but one part
of each questions routine and simple (though comprehensive), and the
remaining part possibly requiring some creativity. The material taught
in this class was important and significant, and thus the main purpose
of the exam, as I see it, is to encourage you to review that material
and to verify that you've understood the main points. Ergo the emphasis
on the comprehensive though routine and simple; the bit about
creativity will have relatively small overall weight and will be there
mostly to make happy those of you who really do need a challenge.
The exam will take place on Tuesday July 17, 2001, at 13:30 at Ulam
Canada. Because of some travel plans, I will not be present, but Liat
Kessler will be there to answer your questions. (Moed B is on September
9, 2001, at 10:00 at Mathematics 110 and I will be present). I remind
you that the final grade will be a weighted average of the final exam
grade f and the homework grade h, with weights
0.85f+0.15h if h>f and
0.93f+0.07h if h<f (i.e., the
carrot has been bigger than the stick).
In principle, all that was discussed in class or in the problem
session or in the homework sheets is fair game. Topics that we
discussed with precision, I expect you to know with precision. Topics
on which I gave no details (and didn't indicate the details as "easy
homework exercises"), you are only supposed to know to the same level
of detail that I provided. For your convenience, here is a list of the
topics covered in the course:
- The area function of parallelograms in the plane and its basic
- Signed area in the plane and its basic properties.
- Definition of volume forms in general.
- Anti-symmetry of volume forms.
- Planar area and 2x2 determinants.
- Uniqueness (up to a scalar) of 2-dimensional volume forms.
- Permutations in two notations.
- Composition and inverses of permutations.
- Every permutation is a product of transpositions.
- The sign of a permutation.
- The sign of a composition and of an inverse.
- Total antisymmetry (behavior under general permutations) of volume
- The "fundamental formula" for a volume form evaluated on an arbitrary
- Definition of the determinant.
- Determinants of 1x1, 2x2 and 3x3 matrices.
- A volume form that vanishes on a basis is identically zero.
- Any two volume forms are proportional.
- Multi-linearity of determinants.
- Equal rows --> the determinant vanishes.
- Volume forms exist on an arbitrary finite dimensional vector space.
- Detecting bases with a volume form.
- Detecting invertibility (and linear independence of rows) with
- Defining determinants using rook arrangements.
- The determinant of the transpose matrix.
- Naive computational complexity of determinants.
- Exchanging rows/columns and the effect on the determinant.
- Adding one row/column to another and the effect on the determinant.
- Order n3 computation of the determinant.
- Determinants of triangular matrices.
- The determinant of a scalar times a matrix.
- Nothing about the determinant of a sum of two matrices.
- The determinant of a product of two matrices.
- A lemma about the sign of a rook arrangement with one rook removed.
- Row- and column-expansion of determinants.
- The adjoint matrix and its relationship with the inverse matrix.
- Cramer's law.
- Reproduction laws of rabbits.
- Translation of the rabbits problem into matrix language.
- Powers of diagonal matrices.
- Powers of conjugate matrices.
- Eigenvalues and eigenvectors.
- A basis of eigenvectors --> diagonalization.
- The characteristic polynomial and finding eigenvalues.
- Finding eigenvectors.
- An algorithm for diagonalization.
- Eigenvectors corresponding to different eigenvalues are
- Complex eigenvalues and eigenvectors.
Powers of the matrix
- The Cayley-Hamilton theorem: a wrong proof, a proof at 2x2, no general
proof in this class.
- The case of repeated eigenvalues (canonical form is C or
- Systems of linear ordinary differential equations.
- Solutions in the one equation case and in the trigonometric and
- Definition of matrix exponentiation.
- Convergence of matrix exponentiation.
- Exponentiation of diagonal matrices.
- Exponentiation of C and of tC.
- Exponentiation of arbitrary matrices by finding their canonical form.
- Differentiating matrices and vectors that depend on a parameter.
- The derivative of an exponential in the matrix case.
- Jordan's theorem.
- Corollary: n different eigenvalues --> diagonalizable.
- Corollary: the 2x2 case.
- Corollary: powers of arbitrary matrices.
- The characteristic polynomial of conjugate matrices.
- Finding the Jordan canonical form.
- Inner products in the Eucledean plane, lengths and angles.
- A formula for the inner product in the Eucledean plane.
- Symmetry, bilinearity and positivity of the inner product in the
- Definition of inner products in general.
- Examples: R2, Rn, spaces of
- Complex numbers.
- Complex conjugation and its behavior under addition and
- Complex conjugation and absolute values.
- Definition of inner products over complex vector spaces (sesquilinearity).
- Examples: Cn, spaces of functions.
- The norm and the cosine of the angle in general inner product spaces.
- The cosines theorem.
- Orthogonality and Pythagoras' theorem.
- The Cauchy-Schwartz lemma (and its proof!).
- The triangle inequality.
- Definition of orthonormal bases.
- Example: the standard basis of Rn or
- Example: eikt on C[0,2π].
- Expanding an arbitrary vector in terms of an orthonormal basis.
A word on Fourier analysis.
- Computing the norm and inner products for vectors given in terms of an
- The Gram-Schmidt orthogonalization procedure.
- The word "isomorphism".
- Corollary (of Gram-Schmidt): every finite dimensional complex
inner product space is isomorphic to Cn (real spaces to
- Orthogonal matrices and orthonormal bases.
- Symmetric matrices and the statement of the orthogonal
- How to perform orthogonal diagonalization.
- Using orthogonal diagonalization to studying extremas of
- The relation between symmetric matrices and inner products.
- Eigenvectors corresponding to different eigenvalues of a symmetric
matrix are orthogonal.
- Eigenvalues of symmetric matrices are real.
- The orthogonal complement of a subspace.
- Proof of the orthogonal diagonalization theorem.
- Translation of the tomography problem into a linear algebra problem.
- Time estimate for the naive solution.
- The ART (Algebraic Reconstruction Technique) algorithm.
- Time estimate for the ART algorithm.