Standard way to describe vector is using coordinates $\begin{bmatrix} x \\ y \end{bmatrix}$. each coordinate is scalar to stretches or squishes basis vectors, normally $\mathbf{\hat{i}}$ and $\mathbf{\hat{j}}$ which are standard basis vectors. In this section, think about the idea of using different set of basis vectors. Let's define new basis vectors $\mathbf{\vec{b_1}}$ and $\mathbf{\vec{b_2}}..

In this section, let's view "Cramer's rule" by geometrically. Cramer's rule is not the best way to compute solutions of systems of linear equations. Gaussian elimination will always be faster. But understanding Cramer's rule geometrically will help consolidate ideas of relation between determinant and system of linear equations. Example In this setup, we define system of linear equations with 1)..

3-D cross product formula: $$ \mathbf{\vec{v}} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} ; \mathbf{\vec{w}} = \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix} $$ $$ \mathbf{\vec{v}} \times \mathbf{\vec{w}} = \det \begin{pmatrix} \begin{bmatrix} \mathbf{\hat{i}} & v_1 & w_1 \\ \mathbf{\hat{j}} & v_2 & w_2 \\ \mathbf{\hat{k}} & v_3 & w_3 \\ \end{bmatrix} \end{pmatrix} = \mathbf{\hat{i}}(v_2..

2 dimension The cross product of $\mathbf{\vec{v}}$ and $\mathbf{\vec{w}}$, written $\mathbf{\vec{v}} \times \mathbf{\vec{w}}$, is the area of the parallelogram defined by $\mathbf{\vec{v}}$ and $\mathbf{\vec{w}}$. Cross product also considers orientation. 1). If $\mathbf{\vec{v}}$ is on the right of $\mathbf{\vec{w}}$, then $\mathbf{\vec{v}} \times \mathbf{\vec{w}}$ is positive and equal to the..

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 \\ ..