Wk4

Introduction: Einstein summation convention and the symmetry of the dot product

$$AB = C \newline C_{ik} = a_{ij}b_{jk}$$

Multiplying a 2x3 matrix with a 3x4 matrix results in a 2x4 matrix; by first row times each columns

$$\begin{pmatrix}u_i\\\vdots\end{pmatrix}\cdot\begin{pmatrix}v_i\\\vdots\end{pmatrix}$$

$$[u_1 u_2 ... u_n]\begin{bmatrix}v_1\\v_2\\\vdots\\v_n\end{bmatrix}$$

$$\begin{pmatrix}a_{11} \cdots a_{1n}\\\vdots\\a_{n1} \cdots a_{nn}\end{pmatrix} \begin{pmatrix}b_{11} \cdots b_{1n}\\\vdots\\b_{n1} \cdots b_{nn}\end{pmatrix}$$

$$(ab)_{23} = a_{21}b_{13} + a_{22}b_{23} + \cdots + a_{2n}b_{n3} \newline ab_{ik} = \sum_{j} a_{ij}b_{jk} = a_{ij}b_{jk}$$

Matrices changing basis

NOTE: Projection only works if orthogonal unit space

Doing a transformation in a changed basis

Bears basis had a first axis of three, one and a second axis of one, one. Let's say I have a vector x, y defined in bears basis. As I want to transform it by doing something like a rotation of 45 degrees.

$$\frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}$$

Orthogonal matrices

$$A_{ij}^T = A_{ji} \newline \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}^T = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$$

The Gram–Schmidt process

Example: Reflecting in a plane