June 2019 - Infinity is Really Big

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 ${\Bbb R}^{2}$ .

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

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