How to find the standard matrix of a linear transformation?
Understand the Problem
The question is asking how to determine the standard matrix associated with a given linear transformation. This involves understanding the transformation's effects on basis vectors and representing those results in matrix form.
Answer
The standard matrix \( A \) associated with the linear transformation is expressed as: $$ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} $$
Answer for screen readers
The standard matrix ( A ) associated with the linear transformation ( T ) can be represented as:
$$ A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} $$
Steps to Solve
- Identify the Linear Transformation
Define the linear transformation ( T ) you are working with. For example, let's say ( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 ) is given by ( T(\mathbf{x}) = A \mathbf{x} ), where ( A ) is a matrix that describes the transformation.
- Apply the Transformation to Basis Vectors
Apply the transformation ( T ) to the standard basis vectors of ( \mathbb{R}^2 ), which are ( \mathbf{e_1} = \begin{bmatrix} 1 \ 0 \end{bmatrix} ) and ( \mathbf{e_2} = \begin{bmatrix} 0 \ 1 \end{bmatrix} ).
- Calculate the Images
Find the images of these basis vectors under the transformation:
For ( \mathbf{e_1} ): $$ T(\mathbf{e_1}) = A \begin{bmatrix} 1 \ 0 \end{bmatrix} = \begin{bmatrix} a_{11} \ a_{21} \end{bmatrix} $$
For ( \mathbf{e_2} ): $$ T(\mathbf{e_2}) = A \begin{bmatrix} 0 \ 1 \end{bmatrix} = \begin{bmatrix} a_{12} \ a_{22} \end{bmatrix} $$
- Form the Standard Matrix
The standard matrix ( A ) associated with ( T ) can then be constructed using the images of the basis vectors as columns:
$$ A = \begin{bmatrix} T(\mathbf{e_1}) & T(\mathbf{e_2}) \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} $$
- Interpreting the Standard Matrix
This matrix ( A ) can now be used to apply the transformation ( T ) to any vector ( \mathbf{x} ) in ( \mathbb{R}^2 ) by multiplying the matrix with that vector:
$$ T(\mathbf{x}) = A \mathbf{x} $$
The standard matrix ( A ) associated with the linear transformation ( T ) can be represented as:
$$ A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} $$
More Information
The standard matrix for a linear transformation provides a concise way to describe how the transformation acts on any vector in the vector space. Knowing how the transformation interacts with the basis vectors gives us all the information we need to understand the transformation overall.
Tips
- Forgetting to apply the transformation to both basis vectors.
- Confusing the place of the basis vectors in the resulting matrix; they should always form the columns.
- Not checking that the transformation is linear before proceeding with matrix construction.
