(c) Determine the concatenated transformation matrix for translation by vector [1 1] followed by rotation of 45 degrees in 2D square matrix of points given [-1,-1] [1,1] [1,-1] [-1,1]

Sure, here are the steps to determine the concatenated transformation matrix for translation by vector [1 1] followed by rotation of 45 degrees in 2D square matrix of points given [-1,-1] [1,1] [1,-1] [-1,1]:

1. First, we need to create the translation matrix. In 2D, a translation matrix for a vector [x y] is given by:

   T = [1 0 x] [0 1 y] [0 0 1]

   So for a translation vector of [1 1], the translation matrix T is:

   T = [1 0 1] [0 1 1] [0 0 1]

2. Next, we need to create the rotation matrix. In 2D, a rotation matrix for an angle θ is given by:

   R = [cos(θ) -sin(θ) 0] [sin(θ) cos(θ) 0] [0 0 1]

   So for a rotation of 45 degrees, the rotation matrix R is:

   R = [cos(45) -sin(45) 0] [sin(45) cos(45) 0] [0 0 1]

3. Now, we need to concatenate the translation and rotation matrices. This is done by matrix multiplication. The order of multiplication is important. Since we are first translating and then rotating, we multiply the rotation matrix R by the translation matrix T:

   RT = R * T

4. Finally, we apply this concatenated transformation matrix to each of the points in the 2D square matrix. This is done by multiplying the transformation matrix by each point vector:

   p' = RT * p

   where p is the original point vector and p' is the transformed point vector.

Remember that these calculations require the points to be in homogeneous coordinates. That means each 2D point [x y] needs to be represented as a 3D point [x y 1] before the transformation. After the transformation, you can convert the point back to 2D by removing the third coordinate.