Finding the transformation matrix

Finding the transformation matrix

We are always given the transformation matrix to transform shapes and vectors, but how do we actually give the transformation matrix in the first place? To do this, we must take a look at two unit vectors. With each unit vector, we will imagine how they will be transformed. Then take the two transformed vector, and merged them into a matrix. That matrix will be the transformation matrix. We will first examine the different types of transformations we will encounter, and then learn how to find the transformation matrix when given a graph.

Lessons

In this section, we will be focusing on finding the transformation matrix.
Given a picture or a description of the transformation, how do we find the transformation matrix? What we do is to take a look at the two unit vectors:
two unit vectors

We want to ask ourselves how the transformation given in the question changes these two unit vectors.
Let’s say that the unit vector unit vector 1 and 0 transforms from unit vector 1 and 0 to unit vector a and b and the unit vector unit vector 0 and 1 transforms from unit vector 0 and 1 to unit vector c and d. Then we say that:
transformation of unit vectors

Then we combine these two column vectors into one matrix.
Hence, the transformation matrix is:
transformation matrix a, b ,c ,d
  • Introduction
    Finding the transformation matrix overview

  • 1.
    Transformation of vectors
    You are given a vector and a description of the transformation. Determine the new vector when it is transformed, and graph them:
    a)
    Finding the transformation matrix, scaled by a factor of 4

    b)
    Finding the transformation matrix, scaled by a factor of 12\frac{1}{2}

    c)
    Finding the transformation matrix, rotated 90° counter-clockwise

    d)
    Finding the transformation matrix, rotated 270° clockwise

    e)
    Finding the transformation matrix, reflected on the xx-axis

    f)
    Finding the transformation matrix, reflected on the yy-axis


  • 2.
    Finding the transformation matrix
    You are given a picture of a transformation taking place. Find the matrix that causes this transformation:
    a)
    Finding the transformation matrix

    b)
    scaled by a factor of 13\frac{1}{3}
    Finding the matrix causing the transformation matrix

    c)
    How a matrix causes transformation

    d)
    Finding the transformation matrix