Finding the Matrix for Linear Transformations with Specific Properties

Introduction

When dealing with linear transformations, one of the fundamental questions is how to find the matrix that represents a specific transformation. This article will delve into finding the matrix for a linear transformation that satisfies certain conditions, particularly focusing on the properties of the transformation matrix with respect to the standard basis.

Understanding Linear Transformations and Bases

To speak about the matrix of a linear transformation, it is crucial to specify bases for both the domain and the codomain. This article assumes a linear transformation ( T: mathbb{R}^3 to mathbb{R}^3 ). The standard basis for (mathbb{R}^3) consists of the coordinate vectors ( mathbf{e}_1, mathbf{e}_2, mathbf{e}_3 ).

Constructing the Transformation Matrix

Consider a specific linear transformation ( T ) such that ( T(mathbf{e}_1 - mathbf{e}_3) mathbf{u} ), ( T(mathbf{e}_1) mathbf{v} ), and ( T(mathbf{e}_2) mathbf{w} ). Here, ( mathbf{u} begin{pmatrix} 1 0 -1 end{pmatrix} ) and ( mathbf{v} begin{pmatrix} 6 -1 1 end{pmatrix} ). Since ( mathbf{u} ) and ( mathbf{v} ) are linearly independent, they can be extended to a basis of (mathbb{R}^3) by adding a third vector ( mathbf{w} ) that is not in the span of ( mathbf{u} ) and ( mathbf{v} ). This means that ( mathbf{w} begin{pmatrix} w_1 w_2 w_3 end{pmatrix} ) where ( w_1 cdot 7 w_2 cdot w_3 eq 0 ).

Determining the Transformation Matrix

The matrix of ( T ) with respect to the basis ( { mathbf{e}_1 - mathbf{e}_3, mathbf{e}_1, mathbf{e}_2 } ) is given by:

[ begin{pmatrix} v w v - u end{pmatrix} ]

Here, ( v ) and ( w ) are arbitrary vectors in (mathbb{R}^3). If we set ( v begin{pmatrix} x_1 x_2 x_3 end{pmatrix} ) and ( w begin{pmatrix} y_1 y_2 y_3 end{pmatrix} ), the matrix becomes:

[ begin{pmatrix} x_1 y_1 x_1 - 6 x_2 y_2 x_2 - 1 x_3 y_3 x_3 - 1 end{pmatrix} ]

Converting to the Standard Basis

To find the matrix of ( T ) with respect to the standard basis, we need to conjugate the above matrix by the transition matrix. The transition matrix from the standard basis to the given basis is:

[ begin{pmatrix} 1 6 w_1 0 -1 w_2 -1 1 w_3 end{pmatrix} ]

The matrix of ( T ) with respect to the standard basis is then:

[ [T]_{mathcal{S}} frac{1}{D} begin{pmatrix} 1 6 w_1 0 -1 w_2 -1 1 w_3 end{pmatrix} begin{pmatrix} x_1 y_1 x_1 - 6 x_2 y_2 x_2 - 1 x_3 y_3 x_3 - 1 end{pmatrix} begin{pmatrix} w_2 -w_1w_3 w_2 - w_1w_3 6w_3 - w_1 1 7 end{pmatrix} ]

where ( D w_1 cdot 7 w_2 cdot w_3 eq 0 ). This represents a complete description of the transformation, showing the huge variety of possible transformation matrices.

Example of a Transformation Matrix

For a more concrete example, let's set ( a s ), ( b t ), and ( w_3 1 ), with all other variables equal to zero. The matrix simplifies to:

[ begin{pmatrix} 6t -42t 36 6t -t -7t 6 -t t 7t - 6 t end{pmatrix} ]

This matrix represents a transformation that satisfies the condition ( T(mathbf{u}) mathbf{v} ) and also ( T(mathbf{v}) smathbf{u} ) and ( T(mathbf{w}) tmathbf{v} ).

Conclusion

In summary, the matrix for a linear transformation can be quite complex, especially when specific conditions are imposed. The process involves specifying bases for the domain and codomain, setting up the transformation using the standard basis, and then converting the matrix using the transition matrix. The variety of possible matrices depends on the freedom in choosing the vectors in the basis.