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 <img src='https://s0.wp.com/latex.php?latex=2%5Ctimes+2&bg=ffffff&fg=000000&s=0' alt='2\times 2' title='2\times 2' class='latex' /> 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.

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

… Read the rest

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 $180^{\circ}$ .

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

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 25, 2019September 12, 2019

Linear Algebra in Three Dimensions

Visual Linear Algebra Online, Section 1.6

The solution set of a certain system of three linear equations and three unknowns is the line of intersection of three planes. ‘Typically’, three planes will intersect at just one point, however.

Linear algebra has immense practical value in our modern world. Statistics, business, science, and engineering all make heavy use of the

… 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 ${\Bbb R}^{2}$ .*

*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

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 $y=x$ .

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 30, 2019September 5, 2019

Systems of Linear Equations in Two Dimensions

Visual Linear Algebra Online, Section 1.3

*Visualizing the parametric vector form of the solution set of a system with infinitely many solutions.*

Jack and Jill earn a combined $79.50 when Jack works 2 hours and Jill works 3 hours. On the other hand, if Jack works 3 hours and Jill works 2 hours, they make $78.75 total. What are

… Read the rest

May 21, 2019September 5, 2019

Vectors in Two Dimensions

Visual Linear Algebra Online, Section 1.2

A big-league baseball player smashes a four-seam fastball with the sweet-spot of the bat. Is it a home run? The answer is — it depends. It depends on the launch angle and on it being a “fair ball”. In other words, what is the

… Read the rest

May 17, 2019May 17, 2019

Linear and Trigonometric Force of Mortality

Studying for Exam LTAM, Part 1.11

Survival function for a continuous lifetime random random variable with a trigonometric (sinusoidal) force of mortality.

In the last post, “Transformed Power Function Survival Models”, we explored the consequences of assuming certain forms of the survival function for a continuous survival random variable $T_{x}$ .

Here we change our perspective. Instead of starting with … Read the rest

Visual Linear Algebra Online, Section 1.10

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

Visual Linear Algebra Online, Section 1.9

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

Visual Linear Algebra Online, Section 1.8

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.

Visual Linear Algebra Online, Section 1.7

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.

Visual Linear Algebra Online, Section 1.6

Linear algebra has immense practical value in our modern world. Statistics, business, science, and engineering all make heavy use of the

Visual Linear Algebra Online, Section 1.5

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.

Visual Linear Algebra Online, Section 1.4

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

Visual Linear Algebra Online, Section 1.3

Jack and Jill earn a combined $79.50 when Jack works 2 hours and Jill works 3 hours. On the other hand, if Jack works 3 hours and Jill works 2 hours, they make $78.75 total. What are

Visual Linear Algebra Online, Section 1.2

A big-league baseball player smashes a four-seam fastball with the sweet-spot of the bat. Is it a home run? The answer is — it depends. It depends on the launch angle and on it being a “fair ball”. In other words, what is the

Studying for Exam LTAM, Part 1.11

In the last post, “Transformed Power Function Survival Models”, we explored the consequences of assuming certain forms of the survival function for a continuous survival random variable .

In the last post, “Transformed Power Function Survival Models”, we explored the consequences of assuming certain forms of the survival function for a continuous survival random variable $T_{x}$ .