Notes
Lecture 30: Matrices of bilinear forms and orthogonal maps. Notes
Unfortunately, I forgot to record this lecture. I annotated the notes
a bit (in blue).
In the first part, I was trying to explain that the choice of a basis
$(v_1,\ldots,v_n)$ in the $\mathbb{F}$-vector space $V$ sets up a
one-to-one correspondence between $n\times n$ matrices with
coefficients in $\mathbb{F}$ and bilinear forms $V\times V\to
\mathbb{F}$. We see an instance of this in Problem 8.2, where an
inner product on $\mathbb{R}^3$ is defined in terms of a matrix.
In the second part, I went over orthogonal transformations following Jänich.
Lecture 31: Orthogonal and self-adjoint endormorphisms of Euclidean
vector spaces.
Notes
Lecture 32: Self-adjoint operators and the spectral theorem.
Notes
Lecture 33: Quadratic forms and the Principal Axes Theorem.
Somehow, only the audo got recorded. I am sorry.
Notes
Lecture 34: Equivalence relations: row equivalence, the rank theorem,
similarity, Jordan canonical form.
Notes
Lecture 35: More on Jordan Canonical Form. Sylvester Theorem.
Notes
Lecture 36: More on the Sylvester Inertia Theorem.