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Finding the transformation matrix

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Intros
Lessons
  1. Finding the transformation matrix overview
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Examples
Lessons
  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:
    1. Finding the transformation matrix, scaled by a factor of 4
    2. Finding the transformation matrix, scaled by a factor of 12\frac{1}{2}
    3. Finding the transformation matrix, rotated 90° counter-clockwise
    4. Finding the transformation matrix, rotated 270° clockwise
    5. Finding the transformation matrix, reflected on the xx-axis
    6. 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:
    1. Finding the transformation matrix
    2. scaled by a factor of 13\frac{1}{3}
      Finding the matrix causing the transformation matrix
    3. How a matrix causes transformation
    4. Finding the transformation matrix
Topic Notes
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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.
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
Basic Concepts
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