Linear Transformations and Standard Bases in \(R^3\)

A linear transformation is a fundamental concept in linear algebra that maps vectors from one vector space to another while preserving the operations of vector addition and scalar multiplication. In this particular problem, we are tasked with finding the standard matrix of a linear transformation \(T\) that operates from \(R^3\) to \(R^3\).

The standard ordered basis for \(R^3\) consists of a specific set of three orthogonal unit vectors. These vectors are linearly independent and can be used to represent any vector in \(R^3\). They are:

• \(e_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\)
• \(e_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\)
• \(e_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\)

The standard matrix of a linear transformation \(T: R^n \rightarrow R^m\) is constructed by taking the images of the standard basis vectors of \(R^n\) under \(T\) as its columns. Each column will be a vector in \(R^m\).

Calculating Images of Standard Basis Vectors under \(T\)

The problem provides us with how the linear transformation \(T\) acts on each of the standard basis vectors:

• \(T(e_1) = 7e_1 - 5e_3\)
• \(T(e_2) = -2e_2 + 9e_3\)
• \(T(e_3) = e_1 + e_2 + e_3\)

To find the standard matrix, we need to express these transformed vectors as column vectors in \(R^3\):

1. Image of \(e_1\):
   Given \(T(e_1) = 7e_1 - 5e_3\). We can write this as \(T(e_1) = 7e_1 + 0e_2 - 5e_3\).
   Substituting the coordinate representations of \(e_1, e_2, e_3\):
   \(T(e_1) = 7\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} + 0\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} - 5\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 7 \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ -5 \end{pmatrix} = \begin{pmatrix} 7 \\ 0 \\ -5 \end{pmatrix}\)
2. Image of \(e_2\):
   Given \(T(e_2) = -2e_2 + 9e_3\). We can write this as \(T(e_2) = 0e_1 - 2e_2 + 9e_3\).
   Substituting the coordinate representations of \(e_1, e_2, e_3\):
   \(T(e_2) = 0\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} - 2\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} + 9\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ -2 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ 9 \end{pmatrix} = \begin{pmatrix} 0 \\ -2 \\ 9 \end{pmatrix}\)
3. Image of \(e_3\):
   Given \(T(e_3) = e_1 + e_2 + e_3\). We can write this as \(T(e_3) = 1e_1 + 1e_2 + 1e_3\).
   Substituting the coordinate representations of \(e_1, e_2, e_3\):
   \(T(e_3) = 1\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} + 1\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} + 1\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\)

Constructing the Standard Matrix of Linear Transformation \(T\)

The standard matrix of \(T\), often denoted as \([T]\), is formed by arranging these computed column vectors side-by-side. The first column of the matrix will be \(T(e_1)\), the second column will be \(T(e_2)\), and the third column will be \(T(e_3)\).

Combining these column vectors, the standard matrix of \(T\) is:

\([T] = \left( {\begin{array}{} 7&0&1\\ 0&{ - 2}&1\\ { - 5}&9&1 \end{array}} \right)\)

This matrix accurately represents the given linear transformation with respect to the standard basis of \(R^3\).