Linear Transformations

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Abstract

To show a transformation is linear:

  • Check that

To find the standard matrix:

  • If you are able to find a matrix/are given a matrix already, multiply that by the identity matrix
  • Otherwise, apply the transformation to each standard basis vector as needed
    I.e find for the number of basis vectors that is required.

Recall that a matrix is actually a "rule" for a linear transformation of a vector, and recall the definition of a function.

In calculus, is focused on. However, in linear algebra, we focus on .

Properties

The properties of a linear transformation are the same as the properties of a subspace, adapted for linear transformations. See operations on linear transformations for how that would be achieved.

Matrix Transformation

Definition

For , the function defined by for all is called the matrix transformation corresponding to .

We call the domain of and the codomain of .
We say that maps to , and say that is the image of under .

Intuition

Recall that the columns of a matrix tell us where the basis vectors land.

Non Square Matrix Transformation

Definition

For , we have that . See matrix-vector product to find out why this is the case.

Visualization

The matrix transformation:

Maps

Change of Space

Theorem

If any vectors (columns) of a matrix are linearly dependent on the others, the transformation squishes the inputs into a smaller space.

Intuition

Since running Gauss-Jordan Elimination reduces any vectors which are linear combinations to the zero vector, the matrix becomes non-square.

Linear Transformation or Operator

Definition

A function is called a linear transformation (or linear mapping) if for every and for every , we have:

For , a linear transformation is often called a linear operator on

Properties

All linear transformations satisfy the following properties:

  1. All lines stay lines
    • Technically:
  2. The origin stays in place
    • Technically:
Important

Every matrix transformation is a linear transformation.

Tip

For a linear transformation , if we are given for , when we can compute for any .

In particular, if we have where the basis lands, we can compute where any vector lands.

(see span)

Standard Matrix

Definition

The standard matrix of a linear transformation is the identity matrix multiplied by the matrix of the linear transformation. In fact, a linear transformation is completely defined by how it moves the basis vectors.

We denote the standard basis as .

(see matrix multiplication)

Theorem

If is a linear transformation, then is a matrix transformation with the corresponding matrix

that is, for every .

(note that are standard basis vectors)

Example

Let be nonzero and define by for every . Show that is linear, and then find the standard matrix of with . (see projection).

We first show that is linear. Let and . We have:

so is linear. Now, with :

So

Observation

Projections are linear, and can be computed using a matrix transformation.