# 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

![](/files/-LkAyv51-sYbc0Clpi0m)

![](/files/-LkBVBPUsrd8S7TvcupS)

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.

![](/files/-LkBWZyj8e1RYkzZiGbr)

![](/files/-LkBZFEPDVyB-9bgFnxy)

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

## **Orthogonal matrices**

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

![](/files/-LkBbXUSxesLfQRgk_uZ)

## **The Gram–Schmidt process**

![](/files/-LkBdIDrWH3HUaZsgxZc)

## **Example: Reflecting in a plane**


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