\\\\(
\nonumber
\newcommand{\bevisslut}{$\blacksquare$}
\newenvironment{matr}[1]{\hspace{-.8mm}\begin{bmatrix}\hspace{-1mm}\begin{array}{#1}}{\end{array}\hspace{-1mm}\end{bmatrix}\hspace{-.8mm}}
\newcommand{\transp}{\hspace{-.6mm}^{\top}}
\newcommand{\maengde}[2]{\left\lbrace \hspace{-1mm} \begin{array}{c|c} #1 & #2 \end{array} \hspace{-1mm} \right\rbrace}
\newenvironment{eqnalign}[1]{\begin{equation}\begin{array}{#1}}{\end{array}\end{equation}}
\newcommand{\eqnl}{}
\newcommand{\matind}[3]{{_\mathrm{#1}\mathbf{#2}_\mathrm{#3}}}
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\newcommand{\jac}[2]{{\mathrm{Jacobi}_\mathbf{#1} (#2)}}
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\newcommand{\compl}{\mathbb{C}}
\newcommand{\e}{\mathrm{e}}
\\\\)
Week 6, Long Day: Determinants and Vector Geometry
To every square matrix we define a special number called the determinant. How do you compute this number, and what does the number say about the matrix? Today we will introduce these questions.
We will also be entering the world of vectors in the 2D plane and in 3D space. From highschool you have probably seen vectors defined by their coordinates. Today we will introduce vectors without the need of coordinates. An important feature is the ability to clearly distinguish between a vector and its coordinates: The coordinates depend on the chosen basis!
Our exercises today will start out with the fundamentals of adding vectors and multiplying vectors by numbers. Then they will move into the topic of linear independence. When these concepts are in place, you will be ready to work on the concepts of basis and coordinates.
Today’s Key Concepts
Determinants $\mathrm{det}(\mA)$. The expansion method (expansion along a row or a column). Regular and singular matrices. A vector as a coherent pair of length and direction. Linear combinations. Linear dependence and independence. Basis and coordinates. Standard bases and the standard coordinate systems.
Preparation and Syllabus
Today’s topics are covered by eNote 9 Determinants and eNote 10 Geometric Vectors.
Maple Ressources
Determinant
computes the determinant of a square matrix.
For today we have the following two Maple demos: Determinants and
GeometricVectors
Activity Program $ $
- 10.00 – 12.00: Lecture (aud. 42, b. 303A) (link to streaming)
- 12.30 – 17.00: Group exercises in the study areas (b. 302, bottom floor)
- 13.00 – 16.00: Your teachers are present in the study areas
Group Exercises
- Computation of a Determinant. By Hand
- Determinants and Rank
- Teaser Exercise in which the Identity Matrix Appears
- Determinant Acrobatics
- Vectors: Addition and Multiplication by a Scalar
- Linear Combinations
- Linear Dependence or Independence
- Change of Basis and Coordinates in the Plane
- Change of Basis and Coordinates in 3D Space