matrix of a linear transformation Archives - Infinity is Really Big

August 29, 2019October 2, 2020

The Determinant of a Square Matrix

Visual Linear Algebra Online, Section 1.10

The absolute value of the determinant of a $latex 2\times 2$ matrix gives the area of the parallelogram determined by its column vectors. — The absolute value of the determinant of a $2\times 2$ matrix gives the area of the parallelogram determined by its column vectors.

2\times 2 — The absolute value of the determinant of a $2\times 2$ matrix gives the area of the parallelogram determined by its column vectors.

Mathematical beauty arises in many ways, though for most people, it is typically tied to something visual. The beauty of the more austere realm of abstract mathematics usually takes more time to appreciate

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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

July 1, 2019September 20, 2019

High-Dimensional Linear Algebra

Visual Linear Algebra Online, Section 1.7

Image by Danilo Bueno from Pixabay

Many practical situations that require the use of linear algebra can involve hundreds or even thousands of unknowns. Modeling a national economy, optimizing the purchase of supplies, and managing flight schedules are prominent examples.

A vector quantity representing all the data in such a situation might therefore … Read the rest

June 20, 2019September 12, 2019

Matrices and Linear Transformations in Low Dimensions

Visual Linear Algebra Online, Section 1.5

The null space (kernel) of the linear transformation defined by $T\left(\begin{array}{c} x \\ y \end{array}\right)=2x+y$ is a straight line through the origin in the plane ${\Bbb R}^{2}$ .

T\left(\begin{array}{c} x \\ y \end{array}\right)=2x+y — The null space (kernel) of the linear transformation defined by $T\left(\begin{array}{c} x \\ y \end{array}\right)=2x+y$ is a straight line through the origin in the plane ${\Bbb R}^{2}$ .

In mathematics, a matrix is not a simulated reality, but instead just a plain-old rectangular array of numbers. This does not mean, however, that mathematical matrices are uninteresting.

In fact, we … Read the rest