1. Lecture of Week 1: Introduction to linear systems (Feb 2)

A central point of this week is GaussJordan elimination, introduced in the second
lecture. After introducing ourselves, we look at examples of systems of linear equations
The aim is to illustrate, where such systems could occur and how one could solve them
with 'ad hoc' methods. This involves solving equations by combining equations in
a clever way or to eliminate variables until only one variable is left. We see
examples with no solution, several solutions or exactly one solution.

2. Lecture of Week 1: matrices and GaussJordan elimination (Feb 4)

We rewrite systems of linear equations using matrices and introduce
GaussJordan elimination steps: scaling of rows, swapping rows or subtract
a multiple of one row to an other row. We also see an example,
where one has not only one solution or no solution.
Unlike in multivariable calculus, we distinguish between column vectors
and row vectors. Column vectors are nx1 matrices, and row vectors are $1 \times m$W
matrices. A general n x m matrix has m columns and n rows. The output of GaussJordan
elimination is a matrix rref(A) which is in row reduced echelon form:
the first nonzero entry in each row is 1, called leading 1, every column with
a leading 1 has no other nonzero elements and every row above a row with a leading 1
has a leading 1 to the left. "Leaders want to be first, alone in their column and have
leaders above them to the left".

3. Lecture of Week 1: on solutions of linear systems (Feb 6)

How many solutions does a system of linear equations have? The goal of this lecture
will be to see that there are three possibilities: exactly one solution, no solution
or infinitely many solutions. This can be very well be explained geometrically,
as well as algorithmically from the GaussJordan elimination point of view. We also
mention that one can see a system of linear equations Ax=b in two different ways:
the column picture tells that b = x_{1} v_{1} + ... + x_{n} v_{n}
is a sum of column vectors v_{i} of A, the row picture tells that the dot product of
the row vectors with x are the components w_{1} . x = b_{1} of the vector b.

1. Lecture of Week 2: Linear transformations and their inverses (Feb 9)

This week will provides a link between the geometric and algebraic description of
linear transformations. Linear transformations were introduced formally as transformations
T(x) = A x, where A is a matrix.
We learn how to distinguish between linear and nonlinear, linear
and affine transformations. The transformation T(x) = x+5 for example is not linear because
0 is not mapped to 0. We characterize linear transformations by 3 properties:
(i) T(0) = 0 (ii) T(x+y) = T(x) + T(y) and (iii) T(s x) = s T(x).

2. Lecture of Week 2: Linear transformations in geometry (Feb 13)

Rotations, dilations, projections reflections, rotationdilations
or shears: how are they described algebraically? The main point of this lecture is to see how to
translate forth and back between algebraic and geometric description. The key fact is that the column
vectors of a matrix are the images of the basis vectors. We derive the matrices for
each of the mentioned geometric transformations.

3. Lecture of Week 2: Matrix product and inverse (Feb 15)

The composition of linear transformations corresponds to the product
of matrices. The inverse of a transformation is described by the inverse of the matrix.
We can so treat square matrices similar as numbers: they can be added and multiplied.
Many matrices have inverses. There are only two things to be careful about: the
product of two matrices is not commutative and many nonzero matrices have no inverse.
If we take the product of a nxp matrix with a pxm matrix, we obtain a nxm matrix.
The dot product as a special case of a matrix product between a 1xn matrix and a nx1 matrix. It
produces a 1x1 matrix which is a scalar.

 
1. Lecture of Week 3: Image and kernel (Feb 17)

We define the notion of a linear subspace of ndimensional space and the span
of a set of vectors. This is a preparation for the more abstract definition of
linear spaces appearing later. The main algorithm is the computation of the kernel
and image of a linear transformation using row reduction. The image of a matrix A is spanned
by the columns of A which have a leading 1 in rref(A). The kernel of a matrix A
is parametrized by "free variables", the variables for which there is no leading
1 in rref(A). For a nxn matrix, the kernel is trivial if and only if the matrix
is invertible. The kernel is always nontrivial if the nxm matrix satisfies m>n.
If we have more variables than equations, then either the equation has no solution
or infinitely many.

2. Lecture of Week 3: Basis and linear independence (Feb 19)

With the previously defined "span" and the newly introduced linear independence,
one can define a basis for a linear space. It is a set of vectors which span the
space and are linear independent. The standard basis in R^{n} is an example
of a basis. We show that if we have a basis, then every vector can be uniquely
represented as a linear combination of basis elements.
A typical task is to find the basis of the kernel and the basis for the
image of a linear transformation.

1. Lecture of Week 4: Dimension and linear spaces (Feb 22)

This is a rather abstract week. The concept of abstract linear spaces allows
us to talk also about linear spaces of functions, for example. This will be useful
for applications in differential equations.
We show first that the number of basis elements is independent of the basis. It is called
the dimension. The proof uses that if p vectors are linearly independent and q vectors
span a linear subspace V, then p is less or equal to q.
We see the ranknullity theorem: dim ker(A) + dim im(A) is the
number of columns

2. Lecture of Week 4: Coordinates (Feb 27)

We look at coordinates [x]_{B} of a vector with respect to
an arbitrary basis B. Given a basis B in R^{n},
one defines the matrix S which contains the basis as column vectors.
The coordinate transformation from the standard basis to the new basis is
encoded in this invertible matrix S.
We learn how to write a matrix A in
a new basis: it is B = S^{1} A S. If two matrices satisfy this
relation, then they are called similar.

3. Lecture of Week 4: Linear spaces I (Feb 29)

We generalize the concept of linear subspaces of R^{n} and consider
abstract linear spaces. An example is the space X=C([a,b]) of continuous functions
on the interval [a,b], the space C^{infinity} of all smooth functions on the
real axes or the space P_{5} of polynomials of degree smaller
or equal to 5 or the space M_{2} of 2x2 matrices.

1. Lecture of Week 5: Review for hourly I (Mar 3)

This is review for the first midterm. The plenary review already covered
the main material so that this review can focus on questions, practice exams
or more True/False problems.

2. Lecture of Week 5: Linear spaces II (Mar 5)

In this lecture we continue to study linear spaces.
Since this is a rather abstract subject and it can not hurt to see it again
It is also part of the exam. We can also look at the space of all nxm matrices
as a linear space as well as spaces of solutions of differential equations.
While the topic of linear differential equations will be treated earlier, this
is a preview, how linear algebra enters into differential equations: the solution
spaces of linear differential equations are linear spaces.

3. Lecture of Week 5: orthonormal bases and orthogonal projections (Mar 7)

We review orthogonality between vectors u,v by u.v=0 and define then the
notion of an orthnormal basis, a basis which consists of unit vectors which are
all orthogonal to each other. The orthogonal complement of a linear space V
in a R^{n} is defined the set of all vectors perpendicular to all vectors in V.
It can be found as a kernel of the matrix which contains a basis of V as rows.
We then define orthogonal projection onto a linear space. Given an orthonormal
basis {u_{1}, ... u_{n} } in V, we have a formula for the
orthogonal projection:
P(x) = (u_{1}.x) u_{1} + ... + (u_{n}.x) u_{n}.

1. Lecture of Week 6: GramSchmidt and QR factorization (Mar 10)

Gram Schmidt orthogonalization lead to the QR decomposition
of a matrix. We will look at this process geometrically as well
as algebraically. The QR decomposition of a mxn matrix A produces
a matrix Q with othonormal columns and an upper triangular nxn matrix
R so that A = QR. If A is a nxn matrix, then Q will be called
an orthogonal matrix later.
The GramSchmidt process is useful to obtain an orthonormal basis from
a given basis.

2. Lecture of Week 6: Orthogonal transformations (Mar 12)

After defining the transpose of a matrix, we look at
orthogonal matrices, matrices for which A^{T} A = 1.
Rotations and reflections are examples of orthogonal transformations.

3. Lecture of Week 6: Least squares and data fitting (Mar 14)

This is an important lecture from the application point of view.
We learn how to fit data points with any finite set of functions.
An very special example is to fit a set of data by linear functions.
The process is very general: we write down a linear system of equations
Ax = b
which pretends that all data lie on the curves. Then we find the
least square solution x of this system, which is
x = (A^{T} A)^{1} A^{T}.
This formula can always easily be derived from the fact that bAx
is perpendicular to the image of A.

1. Lecture of Week 7: Determinants I (Mar 17)

The determinant is defined by a permutation definition
det(A) = sum_{p patterns} (1)^{upcrossings(p>} A_{1 p(1)} ... A_{n p(n)}.
It immediately implies the Laplace expansion.
The computation of the determinant can be tedious. A faster method is to use
GaussJordan elimination. To derive this, one has to see what happens under
the three elimination steps: swap, scale and subtract.

2. Lecture of Week 7: Determinants II (Mar 19)

Some techniques: computation of the determinant, the permutation
definition, GaussJordan elimination, by Laplace expansion, for triangular
matrices and for partitioned matrices. Later we will learn how to compute
determinants by computing eigenvalues.

3. Lecture of Week 7: Eigenvalues (Mar 21)

Eigenvalues and eigenvectors are defined relatively late in this
course. It is good to see them in concrete examples like rotations,
reflections, shears. As the book, it is a good idea to motivate
the eigenvalues with a discrete dynamical system problem like the problem
to find the growth rate of the Fibonnacci sequence. Here it
becomes evident, why computing eigenvalues and eigenvectors matter.

SPRING BREAK

Hurrey! We hope you enjoy the break.

1. Lecture of Week 8: Eigenvectors (Mar 31)

After spring break, you might have forgotten about eigenvalues
and need to be reminded about eigenvalues. Computing eigenvectors
relates to the computation of the kernel of a linear transformation.

2. Lecture of Week 8: Diagonalization (Apr 2)

If all eigenvalues of a matrix are different, one can diagonalize
A. We also see that if the eigenvalues are the same, like for the
shear matrix, one can not diagonalize. If the eigenvalues are complex
like for a rotation, one can not diagonalize over the reals. Since
we like diagonalization, we like to include complex numbers from now
on.

3. Lecture of Week 8: Complex eigenvalues (Apr 4)

We start with a very short review on complex numbers in class. Course assistants
will do more to get you up to speed with complex numbers.
The fundamental theorem of algebra says that a polynomial of degree n
has n solutions, when counted with multiplicities.
We express the determinant and trace of a matrix in terms of eigenvalues.
Unlike in the real case, these formulas hold now for any matrix.

1. Lecture of Week 9: Review for second midterm (Apr 7)

We review for the second midterm in section. Since there was a plenary
review for all students covering the theory, one could focus on questions
and see the big picture or discuss some TF problems.

2. Lecture of Week 9: Stability (Apr 9)

We study the stability problem discrete dynamical systems. The absolute value of the
eigenvalues determines the stability of the transformation. If all
eigenvalues are in absolute value smaller than 1, then the origin is
asymptotically stable. Also mention at the case, when the matrix is not
diagonalizable like S/2, where S is the shear and where the shear expansion
competes with the contraction in the diagonal.

3. Lecture of Week 9: Symmetric matrices (Apr 11)

The main point of this lecture is to see that symmetric matrices can be diagonalized.
The key fact is that the eigenvectors of a symmetric matrix are perpendicular to each
other. This implies for example that for a symmetric matrix, kernel and image are
perpendicular to each other. It is enough to see the diagonalization theorem intuitively:
if one perturbs the matrix so that the eigenvalues are different, then one can diagonalize.
We also want to see that diagonalization is not always possible. The shear is the bad
guy. One can also mention without proof the Jordan normal form, which gives the
definite answer to the diagonalization question.

1. Lecture of Week 10: Differential equations I (Apr 14)

We learn to solve linear differential equations by diagonalization.
We discuss linear stability of the origin. Unlike in the discrete time case, where
the absolute value of the eigenvalues mattered, the real part of the eigenvalues
are now important. Its always good to keep in mind the one dimensional case, where these facts
are obvious. The point is that linear algebra allows us to reduce the higher dimensional
case to the onedimensional case.

2. Lecture of Week 10: Differential equations II (Apr 16)

A second lecture is necessary for the important topic of applying linear algebra to
solve differential equations x' = A x. While the central idea is to diagonalize A
and solve y' = D y, where D is the diagonalization, we can do so a bit faster.
Write your initial condition x(0) as a linear combination of eigenvectors
x(0) = a_{1} v_{1} + ... + a_{n} v_{n} and get
x(t) = a_{1} v_{1} e^{l1}+ ... + a_{n} v_{n} e^{ln}
We also look at examples where the eigenvalues l_{1} of the matrix A are complex.
An important case for the later is the harmonic oscillator with and without damping. There
would be many more interesting examples from physics.

3. Lecture of Week 10: Nonlinear systems (Apr 18)

This section is covered in a seperate handout also written
by Otto Bretscher. How can nonlinear systems in two dimensions be analyzed
using linear algebra? The key concepts are finding nullclines, equilibria
and their nature using linearization of the system near the equilibria.
Good examples are competing species systems (as the Murray example in the
handout), predatorpray examples (like the Volterra system) or mechanical
systems.

1. Lecture of Week 11: perators on function spaces (Apr 21)

We study linear maps (operators)
on linear spaces. The main example is the operator D as well as polynomials
of the operator D like D^{2} + D + 1.
The goal is to understand that we can
see solutions of differential equations as kernels of linear operators or
write partial differential equations in the form u_{t} = T(u) where
T is a linear operator.

2. Lecture of Week 11: linear differential operators (Apr 23)

The main goal is to be able to solve linear higher order differential
equations p(D) = g using the operator method. Factor the polynomial
p(D) = (Da_{1}) (Da _{2}) ... (Da_{n})
and invert each linear factor (Da_{i}). This is a general
method which works unconditionally. It allows to put together a
"cookbook method", which describes, how to find the special solution
of the inhomogeneous problem.

3. Lecture of Week 11: inner product spaces (Apr 25)

As a preparation for Fourier theory, we introduce the concept
of an inner product which generalizes the dot product. For 2piperiodic
functions, one can define as the integral of f.g from pi to pi
and divide by pi. It has all the properties we know from the dot
product in finite dimensions. An other example of an inner product on the
space of matrices is = tr(A^{T} B).
We mention that we can now do a lot of the geometry, we did before:
Examples are length, angle, GramSchmidt orthogonalization, projections
reflections, orthogonal transformations, etc.

1. Lecture of Week 12: Fourier theory I (Apr 29)

The expansion of a function with respect to the orthonormal basis
1/2^{1/2},cos(n x),sin(nx) leads to Fourier theory.
A nice example to see how Fourier theory is useful is to derive
the Leibniz series for pi/4.
The main motivation is that the Fourier basis is an orthonormal
eigenbasis to the operator D^{2}. It diagonalizes this operator.
We will then use this for partial differential equations.

2. Lecture of Week 12: Fourier theory II (Apr 31)

Perseval identity is the "Pythagorean theorem". It is useful to
estimate how fast a finite sum approximation converges.
We mention also applications like computations of series by
the Perseval identity or by relating them to a Fourier series.
Nice examples are computations of zeta(2) or zeta(4) using the
Perseval identity.

3. Lecture of Week 12: Partial differential equations (Mai 2)

Linear partial differential equations
u_{t} = p(D) u with an even polynomial p are solved in the same
way as ordinary differential equations: by diagonalization.
The Fourier basis diagonalizes the "matrix" D^{2}
and so the polynomial p(D^{2}). This is much more powerful than the
separation of variable method, which we do not do in this course.
For example, the PDE u_{tt} = u_{xx}  u_{xxxx} + 10 u
can be solved nicely with Fourier as. We can even
solve partial differential equations, where we have a driving force like
u_{tt} = u_{xx}  u + sin(t) or higher dimensional heat
and wave equations, as the homework shows.
