Most of linear transformations stretch or squish space. So measuring how much the space is stretched or squished could be very useful for understanding linear transformation. Determinant Because of linear transformation properties, it's possible to predict "how areas in space change", if you know "how much the area of single unit square changes". Factor representing "how much the area of single ..

Linear transformation is completely determined by where the basis vectors($\mathbf{\hat{i}}$ and $\mathbf{\hat{j}}$) land. Matrix multiplication Let's describe the effect of applying one transformation and then another. The overall effect is a new distinct linear transformation, called composition of the two separate transformations. Like any other linear transformation, it can be described with..

In this chapter, we focus on 1). what linear transformation is and 2). relation between linear transformation and matrix-vector multiplication. Transformation is essentially a fancy word for function. It's something that takes in one vector and spits out another vector. So why use the word transformation instead of function if they mean the same thing? This word suggests a way to view input-outp..

Basis vectors In previous chapter, we describe vector as coordinate, where each coordinate value is location (or movement) of $x$-axis or $y$-axis. There is another view of coordinate value: If there is a pair of number $\begin{bmatrix} v_1 \\ v_2 \end{bmatrix}$ that describe a vector $\mathbf{\vec{v}}$, think each coordinate value as a scalar. In XY-coordinate system, there are two special vect..

Vector is the fundamental building block of linear algebra, so it's worth making sure what exactly a vector is. Perspective of Vector Physics student: vectors are arrows, defined by length and direction, pointing in space. Computer Science student: vectors are ordered lists of numbers. Mathematician student: vector can be anything where there's a sensible notion of 1). adding two vectors and 2)...