Summary (AI Generation)

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Matrix

What is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. For example:

$$\mathbf{v}=\left[\begin{array}{ccccc}{x}_1& x_2& x_3& \cdots & x_n\end{array}\right]$$

Matrices are a fundamental tool in 3D graphics for performing transformations on points and vectors.

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Types of Matrices

1. Identity Matrix:

   • A square matrix with 1s on the diagonal and 0s elsewhere.
   • Example:$$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$
   • Represents no transformation.

2. Zero Matrix:

   • A matrix with all elements as 0.
   • Example:$$\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$$

3. Diagonal Matrix:

   • A matrix where all non-diagonal elements are 0.
   • Example:$$\left[\begin{array}{cc}3& 0\\ 0& 5\end{array}\right]$$

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Matrix Operations

Addition and Subtraction:

• Add or subtract matrices element by element.
• Example:$$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]+\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]=\left[\begin{array}{cc}6& 8\\ 10& 12\end{array}\right]$$

Scalar Multiplication:

• Multiply each element of the matrix by a scalar.
• Example:$$2\times \left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]=\left[\begin{array}{cc}2& 4\\ 6& 8\end{array}\right]$$

Matrix Multiplication:

• Multiply matrices by summing the product of rows and columns.
• Example:$$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\times \left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]=\left[\begin{array}{cc}19& 22\\ 43& 50\end{array}\right]$$

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Determinants and Inverses of Matrices

Determinants:

• A scalar value representing the "volume" scaling of a transformation.
• Example (for a 2x2 matrix):$$\text{det}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=ad-bc$$

Inverses:

• The inverse of a matrix "reverses" its transformation.
• A matrix ( A ) has an inverse ( A^{-1} ) if ( A \times A^{-1} = I ) (Identity matrix).

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Understanding these basics is essential for working with transformations in 3D graphics.

Maxwell's equations:

equation	description
$\mathrm{\nabla }\cdot \overrightarrow{\mathbf{B}}=0$	divergence of $\overrightarrow{\mathbf{B}}$ is zero
$\mathrm{\nabla }\times \overrightarrow{\mathbf{E}}+\frac{1}{c}\frac{\partial \overrightarrow{\mathbf{B}}}{\partial t}=\overrightarrow{\mathbf{0}}$	curl of $\overrightarrow{\mathbf{E}}$ is proportional to the rate of change of $\overrightarrow{\mathbf{B}}$
$\mathrm{\nabla }\times \overrightarrow{\mathbf{B}}-\frac{1}{c}\frac{\partial \overrightarrow{\mathbf{E}}}{\partial t}=\frac{4\pi }{c}\overrightarrow{\mathbf{j}}\mathrm{\nabla }\cdot \overrightarrow{\mathbf{E}}=4\pi \rho$	wha?

Linear Algebra