reflection Archives - Infinity is Really Big

July 29, 2019September 24, 2019

The Inverse Matrix of a Linear Transformation

Visual Linear Algebra Online, Section 1.9

The inverse of a rotation is another rotation by the same angle, but in the opposite direction.

Inverse functions are a kind of high-technology in mathematics. They can help you solve infinitely many problems at once!

In linear algebra, some linear transformations on finite-dimensional Euclidean space ${\Bbb R}^{n}$ have inverse functions. Those that … Read the rest

July 12, 2019October 31, 2019

Matrix Multiplication and Composite Transformations

Visual Linear Algebra Online, Section 1.8

The composition of two reflections in the opposite order produces images (blue) that are rotations of each other by $180^{\circ}$ .

180^{\circ} — The composition of two reflections in the opposite order produces images (blue) that are rotations of each other by $180^{\circ}$ .

Function composition is a fundamental binary operation that arises in all areas of mathematics. Function composition is a useful way to create new functions from simpler pieces.

When the functions are linear transformations… Read the rest

June 12, 2019August 29, 2019

Linear Transformations in Two Dimensions

Visual Linear Algebra Online, Section 1.4

The set $A$ (red) is the butterfly curve and $T(A)$ (blue) is its image under a linear transformation that reflects across the line $y=x$ .

The set $A$ (red) is the butterfly curve and $T(A)$ (blue) is its image under a linear transformation that reflects across the line $y=x$ .

We have already alluded to the idea of linear transformations in two dimensions at various points in the first three sections. One example we encountered, in Section 1.1, “Points, Coordinates, and

… Read the rest

May 7, 2019May 7, 2019

Transformed Power Function Survival Models

Studying for Exam LTAM, Part 1.10

A family of transformed power function survival functions $S_{0}(t)=1-\frac{t^{p}}{\omega^{p}}$ .

S_{0}(t)=1-\frac{t^{p}}{\omega^{p}} — A family of transformed power function survival functions $S_{0}(t)=1-\frac{t^{p}}{\omega^{p}}$ .

A power function is defined by a formula of the form $f(t)=at^{p}$ for some constants $a$ and $p$ .

For certain choices of $a$ and $p$ , such a function could serve as the force of mortality for a continuous survival random variable $T_{0}$ . But it could not serve as … Read the rest