Finding the transformation matrix

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:

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

Then we combine these two column vectors into one matrix.
Hence, the transformation matrix is:

• Introduction
  Finding the transformation matrix overview

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• 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)

  , scaled by a factor of 4
  
  
  b)

  , scaled by a factor of $\frac{1}{2}$
  
  
  c)

  , rotated 90° counter-clockwise
  
  
  d)

  , rotated 270° clockwise
  
  
  e)

  , reflected on the $x$-axis
  
  
  f)

  , reflected on the $y$-axis
  
  

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

  
  
  
  b)

  scaled by a factor of $\frac{1}{3}$
  
  
  
  c)

  
  
  
  d)

  
  
  

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