# Adjoint mapping and duality

- Author:
- Eugene Yablonski

**Active viewpoint (upper picture)**: T is a linear transformation of the plane that maps the red basis vectors to the green ones. Its matrix A=[T] is the same in either basis.

**Passive viewpoint (lower picture)**: Every vector

## Tasks

1. Drag the green points around and observe how the purple vectors move.
2. Find the inner product of each purple vector and each red vector. Conclude that purple vectors are orthogonal to the corresponding red grid lines.
Hint: Observe that and <T*

**v**|**u**>=<**v**|T**u**> ...