# The proof of inequality by mathematical induction

http://mathcentral.uregina.ca/qq/database/qq.09.06/carol1.html

# torus is a surface having genus one

torus is a surface having genus one

# mathematica graphing a 2 by 2 matrix as an object

is there a possibility to “transform” a matrix (inheriting the input-output “weights”) into a graph object inheriting the matrix relations?

you can convert it to an edge-weighted graph using `WeightedAdjacencyGraph`. This will give you a complete graph (a `Graph` expression) in which each vertex is also connected to itself. I am not sure how much sense it makes treat this matrix as a graph, given the full connectivity.

Since you have a fully connected graph, it does not matter in your case, but be aware that `WeightedAdjacencyGraph` represents missing edges with `Infinity`, not with zero. This is somewhat annoying because it is inconsistent with `WeightedAdjacencyMatrix`, which uses `0`. While `Infinity` does make sense in some (not all) applications, it is much more common to see data that uses 0.

# Geometry of Linear Transformations of the Plane – HMC Calculus Tutorial

If V=R2 and W=R2, then T:R2R2 is a linear transformation if and only if there exists a 22 matrix A such that T(v)=Av for all vR2. Matrix A is called the standard matrix for T. The columns of A are T01 and T10 , respectively. Since each linear transformation of the plane has a unique standard matrix, we will identify linear transformations of the plane by their standard matrices. It can be shown that if A is invertible, then the linear transformation defined by A maps parollelograms to parallelograms. We will often illustrate the action of a linear transformation T:R2R2 by looking at the image of a unit square under T