A Bridge To Linear Algebra, Atanasiu D., Mikusiński P., 2019

A Bridge To Linear Algebra, Atanasiu D., Mikusiński P., 2019.

   This book is intended to be read, with or without help from an instructor, as an introduction to the general theory presented in a standard linear algebra course. Students are encouraged to read it before or parallel with a standard linear algebra textbook as a study guide, practice book, or reference source for whatever and whenever they have problems understanding the general theory. This book can also be recommended as a student aid and its material assigned by an instructor as a reference source for students needing some coaching, clarification, or PRACTICE!

A Bridge To Linear Algebra, Atanasiu D., Mikusiński P., 2019

Singular value decomposition of 3x2 matrices.
In Chapter 3 we discussed orthogonal diagonalization and spectral decomposition of 2 x 2 matrices. We proved that a 2 x 2 matrix can be orthogonally diagonalized and has a spectral decomposition if and only if it is symmetric. In this chapter we obtain similar results for 3 x 2 matrices. The method described here generalizes to matrices of any dimension and has numerous practical applications, including, data compression, noise reduction, or data analysis.

We start by observing that for an arbitrary 3x2 matrix A, the 2x2 matrix AT A is symmetric, so it is easier to work with and it gives us useful information about the original matrix A.

1 Basic ideas of linear algebra.
1.1 2x2 matrices.
1.2 Inverse matrices.
1.3 Determinants.
1.4 Diagonalization of 2x2 matrices.
2 Matrices.
2.1 General matrices.
2.2 Gaussian elimination.
2.3 The inverse of a matrix.
3 The vector space R2.
3.1 Vectors in R2.
3.2 The dot product and the projection on a vector line in R2.
3.3 Symmetric 2x2 matrices.
4 The vector space R3.
4.1 Vectors in R3.
4.2 Projections in R3.
5 Determinants and bases in R3.
5.1 The cross product.
5.2 Calculating inverses and determinants of 3x3 matrices.
5.3 Linear dependence of three vectors in R3.
5.4 The dimension of a vector subspace of R3.
6 Singular value decomposition of 3x2 matrices.
7 Diagonalization of 3x3 matrices.
7.1 Eigenvalues and eigenvectors of 3x3 matrices.
7.2 Symmetric 3x3 matrices.
8 Applications to geometry.
8.1 Lines in R2.
8.2 Lines and planes in R3.
9 Rotations.
9.1 Rotations in R2.
9.2 Quadratic forms.
9.3 Rotations in R3.
9.4 Cross product and the right-hand rule.
10 Problems in plane geometry.
10.1 Lines and circles.
10.2 Triangles.
10.3 Geometry and trigonometry.
10.4 Geometry problems fromthe International Mathematical Olympiads.
11 Problems for a computer algebra system.
12 Answers to selected exercises.

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