Linear Transformations

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Linear Transformations

Definition: A transformation (or operator if ) is defined to be linear if the image is comprised of only linear equations for every mapping , that is . For any vectors and any scalar a transformation is linear if and .

Let's first look at an example of a linear transformation. Consider the following linear transformation defined by the following equations:

We note that the equations forming the image, that is , , and are all linear, so this transformation is also considered linear and that . For example, if we take the vector and apply our linear transformation, we obtain a resultant vector , and we say that is the image of under the linear transformation . In general, a linear transformation is generally defined by the following equations:

In matrix notation we can represent this transformation as . is called the standard matrix for the linear transformation , though sometimes we use the notation instead. Either way, the standard matrix is created from the coefficients from the system of linear equations defining the image of .

This linear transformation is defined by the standard matrix , so we say that is multiplication by and often denote it with the notation .

Either way, these transformations will geometrically transform some vector or point in to some other vector or point in .

Properties of Linear Transformations

We've already stated the following two properties in the definition of a linear transformation, but now we will prove their existence.

Property 1: If is a linear transformation, then for any vectors it follows that .
  • Proof: Suppose that is a linear transformation and is multiplication by . Thus it follows that:
Property 2: If is a linear transformation, then for any vector and any scalar it follows that .
  • Proof: Suppose that is a linear transformation and is multiplication by . Thus it follows that:


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