2. Consider two transformations, a horizontal shear given by f(x)=B -11 and a ve

Question

2. Consider two transformations, a horizontal shear given by f(x)=B -11 and a vertical shear given by gli- x. (a) The unit square is transformed under f to a parallelogram. Find the coordinates of the parallelogram and sketch it. The first parallelogram is transformed under g to a second parallelgram. Find the coordinates of the second parallelogram and sketch it. Find the matrix of the composition function gof by multiplying the matrices for f and g in the proper order. Verify that the columns of the resulting matrix are two of the vertices of second parallelogram.

Explanation / Answer

a)

f =[1 -1 ;0 1]

f =

     1    -1
     0     1

A = [ 0 1 1 0 ; 0 0 1 1]

A =

     0     1     1     0
     0     0     1     1

(x,y)
where x are is first row and y are in 2nd column

hence point are each column example (0,0) . (1,0) etc

now

f*A

ans =

     0     1     0    -1
     0     0     1     1

hence coordinates are (0,0),(1,0) ,(0,1),(-1,1)

after applying g

g = [1 0 ; 1 1]

g =

     1     0
     1     1

g*(f*A)

ans =

     0     1     0    -1
     0     1     1     0

points are (0,0),(1,1),(0,1),(-1,0)

g o f

=

g*f

ans =

     1    -1
     1     0

we can verify by

(g*f)*A

ans =

     0     1     0    -1
     0     1     1     0

both are same

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