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How do you find the rule of a linear transformation?

Published in Linear Algebra 4 mins read

Finding the rule of a linear transformation involves determining how the transformation acts on any vector in its domain. Given that a linear transformation T satisfies two key properties as defined in the reference:

  1. T(x + y) = T(x) + T(y) for all x,y in the domain
  2. T(cx) = cT(x) for all x in the domain and scalar c

And knowing that T(0) = 0. We can exploit these properties to define the transformation. The most common method involves finding the images of the basis vectors.

Steps to Determine the Rule of a Linear Transformation

Here’s a breakdown of the steps:

  1. Identify a Basis: Choose a basis for the domain vector space. The standard basis is often the easiest to use. For example, in R2, the standard basis is { (1, 0), (0, 1) }. In R3, it's { (1, 0, 0), (0, 1, 0), (0, 0, 1) }.

  2. Find the Images of the Basis Vectors: Determine where the transformation T maps each of these basis vectors. This information is often provided or can be derived from the problem statement. For example, you might be given T(1,0) = (2,3) and T(0,1) = (-1,1).

  3. Express a General Vector as a Linear Combination: Write a general vector in the domain as a linear combination of the basis vectors. For example, in R2, any vector (x, y) can be written as x(1, 0) + y(0, 1).

  4. Apply the Linear Transformation: Use the properties of linear transformations to find the image of the general vector. This involves applying the transformation T to the linear combination and using the facts that T(x + y) = T(x) + T(y) and T(cx) = cT(x).

  5. Write the Rule: Express the result from step 4 as a function of the components of the general vector. This gives you the rule for the linear transformation.

Example

Let’s say T: R2 → R2 is a linear transformation such that:

  • T(1, 0) = (2, 3)
  • T(0, 1) = (-1, 1)

We want to find T(x, y) for a general vector (x, y) in R2.

  • Step 1: The standard basis is {(1, 0), (0, 1)}.

  • Step 2: We are given T(1, 0) = (2, 3) and T(0, 1) = (-1, 1).

  • Step 3: (x, y) = x(1, 0) + y(0, 1)

  • Step 4: Apply the linear transformation:

    T(x, y) = T[x(1, 0) + y(0, 1)]
    = xT(1, 0) + yT(0, 1) (Using the properties of linear transformations)
    = x(2, 3) + y(-1, 1)
    = (2x, 3x) + (-y, y)
    = (2x - y, 3x + y)

  • Step 5: The rule for the linear transformation is T(x, y) = (2x - y, 3x + y).

Practical Insights

  • Matrix Representation: Every linear transformation from Rn to Rm can be represented by a matrix. The columns of this matrix are the images of the standard basis vectors. In the example above, the matrix would be:

    |  2  -1 |
|  3   1 |

    Multiplying this matrix by the column vector (x, y) gives the same result: (2x - y, 3x + y).

  • Non-Standard Bases: You can use any basis, but the standard basis usually simplifies calculations. If you use a non-standard basis, the process is the same, but you'll need to express the general vector in terms of that basis.

  • Verifying Linearity: After finding the rule, it's a good practice to verify that the resulting transformation is indeed linear by checking if it satisfies the two properties mentioned at the beginning.

By following these steps, you can effectively determine the rule for any linear transformation, given the images of a basis for its domain.

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