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31. Eigen Vectors1 And Linear Transformations2

Dated: 15-06-2025

The Matrix3 of a Linear Transformation2

Let \(\mathbb V\) be an \(n\) dimensional vector space4 and \(\mathbb W\) be an \(m\) dimensional vector space4 and \(T\) be a linear transformation2 from \(\mathbb V\) to \(\mathbb W\).
To associate a matrix3 with \(T\), we choose the bases \(\mathbb B\) and \(\mathbb C\) for \(\mathbb V\) and \(\mathbb W\) respectively.

\[\vec x \in \mathbb V \implies [\vec x]_\mathbb B \in \mathbb R^n \text{ and } [T(\vec x)]_\mathbb C \in \mathbb R^m\]

Let \(\mathbb B = \{\vec b_1, \vec b_2, \ldots, \vec b_n\}\) be the basis for \(\mathbb V\).
If \(\vec x = r_1 \vec b_1 + \cdots + r_n \vec b_n\) then \([\vec x]_\mathbb B = \begin{bmatrix}r_1 \\ \vdots \\ r_n\end{bmatrix}\)
and \(T\) is a linear transformation2

\[T(\vec x) = T(r_1 \vec b_1 + \dots + r_n \vec b_n) = r_1 T(\vec b_1) + \dots + r_n T(\vec b_n)\]
\[\implies [T(\vec x)]_\mathbb C = r_1 [T(\vec b_1)]_\mathbb C + \dots + r_n [T(\vec b_n)]_\mathbb C\]

Since \(\mathbb C\) coordinates are in \(\mathbb R^m\)

\[[T(\vec x)]_\mathbb C = M[\vec x]_\mathbb B\]
\[ M = \begin{bmatrix} [T(\vec b_1)]_\mathbb C & [T(\vec b_2)]_\mathbb C & \cdots & [T(\vec b_n)]_\mathbb C \end{bmatrix} \]

This matrix3 is a matrix3 representation of \(T\) called the matrix3 for \(T\) relative to the bases \(\mathbb B\) and \(\mathbb C\).

Example

Suppose that \(\mathbb B = \{\vec b_1, \vec b_2\}\) is a basis for \(\mathbb V\) and \(\mathbb C = \{\vec c_1, \vec c_2, \vec c_3\}\) is a basis for \(\mathbb W\). Let \(T : \mathbb V \to \mathbb W\) and \(T(\vec b_1) = 3c_1 - 2c_2 + 5c_3\) and \(T(\vec b_2) = 4c_1 + 7c_2 - c_3\).
Find a matrix3 \(M\) for \(T\) relative to \(\mathbb B\) and \(\mathbb C\).

Solution

\[ \because M = \begin{bmatrix} [T(\vec b_1)]_\mathbb C & [T(\vec b_2)]_\mathbb C & \cdots & [T(\vec b_n)]_\mathbb C \end{bmatrix} \]
\[\because T(\vec b_1) = 3c_1 - 2c_2 + 5c_3 \text{ and } T(\vec b_2) = 4c_1 + 7c_2 - c_3\]
\[ \therefore M = \begin{bmatrix} 3 & 4\\ -2 & 7\\ 5 & -1 \end{bmatrix} \]

Linear Transformations2 From \(\mathbb V\) into \(\mathbb V\)

In general, when \(\mathbb W\) is same as \(\mathbb V\) and their basis (i.e. \(\mathbb C\) and \(\mathbb B\)) are same, the matrix3 \(M\) is called matrix3 for \(T\) relative to \(\mathbb B\) or simply, the \(\mathbb B\)-matrix.3
The \(\mathbb B\)-matrix3 for \(T : \mathbb V \to \mathbb V\) satisfies

\[[T(\vec x)]_\mathbb B = [T]_\mathbb B [\vec x]_\mathbb B \quad \forall \vec x \in \mathbb V\]

Linear Transformations2 On \(\mathbb R^n\)

In an applied problem involving \(\mathbb R^n\), a linear transformation2 \(T\) usually appears as matrix3 transformation \(\vec x \to A \vec x\). If \(A\) is diagonalizable5 then there is basis \(\mathbb B\) for \(\mathbb R^n\) consisting of eigen vectors1 of \(A\).

Theorem

Suppose \(A = PDP^{-1}\) where \(D\) is a diagonal6 \(n \times n\) matrix.3 If \(\mathbb B\) is basis for \(\mathbb R^n\) formed from the columns of \(P\) then \(D\) is the \(\mathbb B\)-matrix3 of the transformation2 \(\vec x \to A \vec x\).

References

Read more about notations and symbols.

  1. Read more about eigen values and vectors. ↩↩

  2. Read more about linear transformations. ↩↩↩↩↩↩↩↩

  3. Read more about matrices. ↩↩↩↩↩↩↩↩↩↩↩↩↩

  4. Read more about vector spaces. ↩↩

  5. Read more about diagonization. ↩

  6. Read more about diagonal matrices. ↩