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Table of Contents

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Definition

Identity Matrix is a square matrix with $1$ on the main diagonal and $0$ elements elsewhere.

• Identity matrix is denoted by $I$ and has form where the main diagonal consists of $1$ and all other entries are $0$.
• Any matrix unchanged when multiplied by the identity matrix.

Examples

• $2 {\times }2$ Identity matrix

  $$ I_2 = \begin{bmatrix} \color{red}1 & 0 \\ 0 & \color{red}1 \end{bmatrix} $$

• $3 {\times} 3$ Identity matrix

  $$ I_3 = \begin{bmatrix} \color{red}1 & 0 & 0 \\ 0 & \color{red}1 & 0 \\ 0 & 0 & \color{red}1 \end{bmatrix} $$

• $n {\times} n$ Identity matrix

  

Properties

Multiplicative Identity

$$ AI = IA = A $$

• Identity matrix leaves any matrix unchanged when multiplied by it.

Inverse

$$ I^{-1} = I $$

• Identity matrix is its own inverse.

Transpose

$$ I^{T} = I $$

• The transpose of the indentitty matrix is itself.

Linear Transformation

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Linearity

Transformation preserves the operations of vector addition and scalar multiplication.

The vector resulting from the product of a vector and a scalar is parellel to the original vector and lies on the same line passing through the origin.

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Linear trasformation $T$ must satisfy 2 conditions.

Additivity

$$ T(u+v) = T(u) + T(v) $$

• This means that transforming the sum of 2 vectors($u$ + $v$) is equivalent to transforming each vector individually**($u$, $v$)** and then adding the results.

Homogeneity

$$ T(cu) = cT(u) $$

• This means that transforming a vector scaled by a scalar$(cu)$ is equivalent to scaling the trasformed vector by the scalar ($cT$).

Practice with Desmos

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데스모스 링크

Linear Transformations

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https://www.desmos.com/calculator/tmodmkuybw

Through the graph above, We can intuitively observe the linear transformation.

Parameters

• Polygon $PQRS$
• Transformation matrix $T = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$
  • $i = (a,c)$, $j=(b,d)$
• Vectors $u$, $v$

How to use

• Polygon $PQRS$ can be used to observe the transformation.
• You can control $i$, $j$ to transform the grid by linear trasformation operation.
• As a result, you can see how the polygon transforms!