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09. Linear Transformations

Dated: 17-03-2025

Matrix Equation

An equation of the form \(A \vec x = \vec b\) is a matrix equation in which \(A\) is a matrix1 which acts on a vector2 \(\vec x\) to produce another vector2 \(\vec b\).

Solution of Matrix Equation

Solution of \(A \vec x = \vec b\) consists of those vectors2 \(\vec x\) in the domain that are transformed into \(\vec b\) in the range.

Transformation or Function or Mapping

A transformation \(T\) from \(\mathbb R^n\) to \(\mathbb R^m\) is a rule that assigned to each vector2 in \(\mathbb R^n\), an image vector2 \(T(x)\) in \(\mathbb R^m\).

\[T : \mathbb R^n \to \mathbb R^m\]

The set3 \(\mathbb R^n\) is called the domain of \(T\) and \(\mathbb R^m\) is called the co-domain of \(T\).
For \(x \in \mathbb R^n\) the set3 of all images \(T(x)\) is called the range of \(T\).

The map \(T : \mathbb R^n \to \mathbb R^m\) is a linear transformation if for any two vectors2 \(\vec u, \vec v \in \mathbb R^n\) and the scalars \(c_1, c_2\), following is satisfied

\[T(c_1 \vec u + c_2 \vec v) = c_1 T(\vec u) + c_2 T(\vec v)\]

Example

\[T : \mathbb R^2 \to \mathbb R^2\]
\[T(x, y) = (-x, y)\]
\[T(1, 2) = (-1, 2)\]

\(T\) transforms \(\hat i + 2 \hat j\) to \(- \hat i + 2 \hat j\).

\[ T (\vec v) = A \vec v = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -x \\ y \end{bmatrix} \]

Example

\[ \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \to \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} x_1 \\ x_2 \\ 0 \end{bmatrix} \]

Pasted image 20250317174313.png

\(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{bmatrix}\) projects a \(\vec v \in \mathbb R^3\) to \(\vec v \in \mathbb R^2\) where \(\mathbb R^2 = x_1 \times x_2\).

Linear Transformations

Definition

A transformation or mapping \(T\) is linear if

  • \(T(\vec u + \vec v) = T(\vec u) + T(\vec v)\) for all \(\vec u, \vec v\) in the domain of \(T\).
  • \(T(c \vec u) = c T(\vec u)\) for all \(\vec u\) and all scalars \(c\).

Applications in Engineering

The following is referred to as superposition principle.
Think of \(\vec v_1, \ldots, \vec v_p\) as signals that go into a system or process and \(T(\vec v_1), \ldots, T(\vec v_p)\) as the responses of that system to the signals.
The system satisfies the superposition principle if an input is expressed as a linear combination4 of such signals, the system’s response is the same linear combination4 of the responses to the individual signals.

Note

\(T : \mathbb R^2 \to \mathbb R^2\) such that \(T(\vec x) = r \vec x\). \(T\) is called a contraction when \(0 \le r < 1\) and a dilation when \(r > 1\).

References

Read more about notations and symbols.

  1. Read more about matrices. ↩

  2. Read more about vectors. ↩↩↩↩↩↩

  3. Read more about sets. ↩↩

  4. Read more about linear combinations. ↩↩