# Sage Interactions - Linear Algebra

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Contents

## Numerical instability of the classical Gram-Schmidt 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 Gauss-Jordan method for inverting a matrix

by Hristo Inouzhe

...(goes all the way to invert the matrix)