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# Linear transformation

See how to find the points after a linear transformation
(translation/dilation/reflection/rotation matrix).
7 examples and their solutions.

## Translation Matrix

### Formula

xy + ab = x + ay + b

### Example

△ABC is moved under the translation
(x, y) → (x + 4, y + 1).
Find the vertices of the moved triangle △A'B'C'.
A(2, 1), B(3, 4), C(5, 3)
Solution

## Dilation Matrix

### Formula

kI xy = x'y'
I: identity matrix

### Example

△ABC is moved under the dilation of 2.
Find the vertices of the moved triangle △A'B'C'.
A(2, 1), B(3, 4), C(5, 3)
Solution

## Reflection Matrix: x-axis

100-1 xy = x'y'

### Example

△ABC is moved under the reflection in the x-axis.
Find the vertices of the moved triangle △A'B'C'.
A(2, 1), B(3, 4), C(5, 3)
Solution

## Reflection Matrix: y-axis

-1001 xy = x'y'

### Example

△ABC is moved under the reflection in the y-axis.
Find the vertices of the moved triangle △A'B'C'.
A(2, 1), B(3, 4), C(5, 3)
Solution

## Reflection Matrix: Origin

### Formula

-I xy = x'y'
I: identity matrix

### Example

△ABC is moved under the reflection in the origin.
Find the vertices of the moved triangle △A'B'C'.
A(2, 1), B(3, 4), C(5, 3)
Solution

## Reflection Matrix: y = x

0110 xy = x'y'

### Example

△ABC is moved under the reflection in y = x.
Find the vertices of the moved triangle △A'B'C'.
A(2, 1), B(3, 4), C(5, 3)
Solution

## Rotation Matrix

### Formula

cos θ-sin θsin θcos θ xy = x'y'
Trigonometry (Right Triangle)

### Example

△ABC is moved under the rotation of 60º counterclockwise about the origin.
Find the vertices of the moved triangle △A'B'C'.
A(2, 2), B(4, 2), C(4, 4)
Solution