Standard way Numerically, if $\mathbf{\vec{v}}$ and $\mathbf{\vec{w}}$ are in the same dimension, dot product is 1). pairing all of the coordinates, 2). multiplying those pairs together, and 3). adding the results: $$ \mathbf{\vec{v}} = \begin{bmatrix} a \\ b \\ c \end{bmatrix} \ \mathbf{\vec{w}} = \begin{bmatrix} d \\ e \\ f \end{bmatrix} $$ $$ \begin{bmatrix} a \\ b \\ c \end{bmatrix} \cdot \b..

In this section, let's think about non-square matrices geometrically. Non-square matrics are transformations that transform particular dimensional vectors into other dimensional vectors. 2-D (plane) to 3-D (space) If there is a transformation that takes $\mathbf{\hat{i}}$ to the coordinate $\begin{bmatrix} a \\ b \\ c \end{bmatrix}$ and $\mathbf{\hat{j}}$ to the coordinate $\begin{bmatrix} d \\ ..

Let's think about the usefulness of linear algebra. One of the main reasons that linear algebra is broadly applicable is that it can solve system of linear equations.  "System of linear equations":$$\begin{matrix}ax + by + cz = l \\dx + ey + fz = m \\gx + hy + iz = n \\\end{matrix}$$ Let's package "system of linear equations" into single vector equation, which consists of 1). matrix ($A$) which ..

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..