## Linear Transformations of Points
## InstructionsClick and drag the red and orange arrows to change the matrixA of the linear transformation.(Alternatively, you can set the matrix by using the text boxes below the applet.) You can show or hide points on the screen by using the checkboxes. Drag the points u, v, w... around to see how their images Au, Av, Aw... change.
## Exploration- Move
**u**and**v**around and see how*A***u**and*A***v**change. Confirm that the point*A***u**is really the result of applying the matrix*A*to the point**u**. - Put several points on the screen and place them in a straight line. What do you see?
- Which transformation makes
*A***u**=**u**,*A***v**=**v**,*A***w**=**w**, etc.? - Find a transformation that reflects points across the
*x*-axis. Same for*y*-axis. - Find a transformation that multiplies vectors
by 2. So,
*A***u**= 2**u**,*A***v**= 2**v**, etc. - Find a transformation that projects points
"orthogonally" onto the
*x*-axis. So, for example*A*(3, 5) = (3, 0) and*A*(-13, -7) = (-13, 0). More generally,*A*(*x*,*y*) = (*x*, 0). - In the previous example, what is the range of the transformation? The codomain? What are their dimensions?
- Find a matrix
*A*so that*A***u**always lies somewhere on the line*y*=*x*, no matter where**u**is. - Place points
**u**,**v**,**w**on the screen so that**u**+**v**=**w**. What do you notice about*A***u**,*A***v**,*A***w**? Does this relationship hold for*all*linear transformations? - Place
**u**at the point (1, 0), place**v**at the point (0, 1), and then try a few different transformations. How are*A***u**and*A***v**related to the columns of*A*? Can you prove this relationship?
Marc Renault, Created with GeoGebra |