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How to Find a Linear Transformation?

Published in Linear Algebra 4 mins read

Finding a linear transformation involves determining a function T that satisfies the properties of linearity: additivity and homogeneity. Here's a breakdown of how to do it:

Understanding Linear Transformations

A transformation T: Rn → Rm is linear if and only if it satisfies these two conditions for all vectors x, y in Rn and all scalars k:

  • Additivity: T(x + y) = T(x) + T(y)
  • Homogeneity: T(kx) = kT(x)

Steps to Finding a Linear Transformation

  1. Determine the Domain and Codomain: Identify Rn (domain) and Rm (codomain) of the transformation T. This will tell you the input and output spaces, respectively.

  2. Find the Transformation of Basis Vectors: If you know how T transforms a basis for Rn, you can find T(x) for any vector x in Rn. A standard basis for Rn is often used (e.g., {(1, 0), (0, 1)} for R2; {(1, 0, 0), (0, 1, 0), (0, 0, 1)} for R3). Let e1, e2, ..., en be a basis for Rn. Determine T(e1), T(e2), ..., T(en). These vectors will be in Rm.

  3. Express a General Vector as a Linear Combination of Basis Vectors: Any vector x in Rn can be written as a linear combination of the basis vectors:

    x = *c1e1 + c2e2 + ... + cn**en

    where c1, c2, ..., cn are scalars.

  4. Apply the Linearity Properties: Use the additivity and homogeneity properties of linear transformations to find T(x):

    T(x) = T( *c1e1 + c2e2 + ... + cn**en )

    T(x) = c1 T(e1) + c2 T(e2) + ... + cn T(en)

  5. Write the Transformation in General Form: Substitute the values of T(e1), T(e2), ..., T(en) that you found in step 2 into the equation from step 4. This will give you a general formula for T(x) in terms of the components of x.

  6. Matrix Representation (Optional): A linear transformation T: Rn → Rm can always be represented by an m x n matrix A. The columns of A are the vectors T(e1), T(e2), ..., T(en), where ei are the standard basis vectors for Rn. Then, T(x) = *A*x.

Example

Let's say we want to find a linear transformation T: R2 → R3 such that T(1, 0) = (2, 1, 3) and T(0, 1) = (0, -1, 1).

  1. Domain and Codomain: R2 and R3 are given.

  2. Transformation of Basis Vectors: We are given T(1, 0) = (2, 1, 3) and T(0, 1) = (0, -1, 1).

  3. General Vector: A general vector x in R2 can be written as x = (x1, x2) = x1(1, 0) + x2(0, 1).

  4. Apply Linearity:
    T(x) = T(x1(1, 0) + x2(0, 1))
    T(x) = x1T(1, 0) + x2T(0, 1)
    T(x) = x1(2, 1, 3) + x2(0, -1, 1)

  5. General Form:
    T(x) = (2x1, x1 - x2, 3x1 + x2)

    So, the linear transformation T is defined by T(x1, x2) = (2x1, x1 - x2, 3x1 + x2).

  6. Matrix Representation:
    The matrix A representing this transformation is
    A = | 2 0 |
    | 1 -1 |
    | 3 1 |
    Then T(x) = Ax

Key Considerations

  • Uniqueness: A linear transformation is uniquely determined by its action on a basis.
  • Kernel and Image: Understanding the kernel (null space) and image (range) of a linear transformation can provide valuable insights.
  • Not all transformations are linear: You must verify that the additivity and homogeneity properties hold to confirm that a transformation is linear.

Finding a linear transformation involves leveraging its defining properties and the concept of a basis to express the transformation in a general, usable form.

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