Sage Interactions  Linear Algebra
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Contents

Sage Interactions  Linear Algebra
 Numerical instability of the classical GramSchmidt algorithm
 Equality of det(A) and det(A.tranpose())
 Linear transformations
 Gerschgorin Circle Theorem
 Singular value decomposition
 Discrete Fourier Transform
 The GaussJordan method for inverting a matrix
 Solution of an homogeneous system of linear equations
 Solution of a non homogeneous system of linear equations
Numerical instability of the classical GramSchmidt algorithm
by Marshall Hampton
Equality of det(A) and det(A.tranpose())
by Marshall Hampton
Linear transformations
by Jason Grout
A square matrix defines a linear transformation which rotates and/or scales vectors. In the interact command below, the red vector represents the original vector (v) and the blue vector represents the image w under the linear transformation. You can change the angle and length of v by changing theta and r.
Gerschgorin Circle Theorem
by Marshall Hampton. This animated version requires convert (imagemagick) to be installed, but it can easily be modified to a static version. The animation illustrates the idea behind the stronger version of Gerschgorin's theorem, which says that if the disks around the eigenvalues are disjoint then there is one eigenvalue per disk. The proof is by continuity of the eigenvalues under a homotopy to a diagonal matrix.
Singular value decomposition
by Marshall Hampton
Discrete Fourier Transform
by Marshall Hampton
The GaussJordan method for inverting a matrix
by Hristo Inouzhe
...(goes all the way to invert the matrix)
Solution of an homogeneous system of linear equations
by Pablo Angulo
Coefficients are introduced as a matrix in a single text box. The number of equations and unknowns are arbitrary.
Solution of a non homogeneous system of linear equations
by Pablo Angulo
Coefficients are introduced as a matrix in a single text box, and independent terms as a vector in a separate text box. The number of equations and unknowns are arbitrary.