For each of the following come up with a 2 times 2 matrix th

For each of the following, come up with a 2 times 2 matrix that transforms the plane in the prescribed way. Shears v_1 = [2 3] by a factor of 2 in the v_2 = [1 2] direction. Stretches by a factor of 4 in the w_1 = [5 2] direction and stretched by a factor of 3 in the w_2 = [2 1].

Solution

a>

a shear matrix is an elementary matrix that represents the addition of a multiple of one row or column to another. Such a matrix may be derived by taking the identity matrix and replacing one of the zero elements with a non-zero value.

here we are given two directions v1 and v2.

v1 gets sheared be a factor k= 2 in the v2 direction

so the matrix elements after the shear transformation would be :

v1\' = v1 and v2\' = v2 + kv1

=> v1\' = [2 ]    and v2\' = [1 + 2*2] = [5]

              [ 3]                     [2 + 2*3]     [8]

hence the 2x2 matrix would be = [2 5]

                                                           [3   8]

b> for stretching :

w1 gets transformet into w1\' and w2 gets transformend into w2\' after the stretch.

let the stretch factor be represented by S

=> w1 is stretched by s1 = 4

=> w1\' = s1*w1

w1\' = [4*5] = [20]

          [4*2]     [8 ]

likewise for w2

w2\' = s2*w2 = [3*2] = [6]

                          [3*1]    [3]

hence the 2x2 matrix is = [20 6]

                                               [8    3]