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Mathematics for Physics

260 Linear Transformation

A linear transformation is a mapping of a vector to another vector in a codomain that is used to represent a change in a scalar value.

Note: Also called linear map.

definition [d] (Linear Transformation = Linear Map) From Hubbard and Hubbard Definition 1.3.2: a linear transformation \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) is a mapping such that for all scalars \(a\) and all \(\mathbf{v}, \mathbf{w} \in \mathbb{R}^{n}\),

  • \(T(\mathbf{v} + \mathbf{w}) = T(\mathbf{v}) + T(\mathbf{w})\)
  • \(T(a\mathbf{v}) = a\, T(\mathbf{v})\) .

where

  • \(T\) is the linear transformation.
  • \(\mathbb{R}^{n}\) is the domain.
  • \(\mathbb{R}^{m}\) is the codomain.
  • \(a\) is a scalar.
  • \(\mathbf{v}, \mathbf{w}\) are vectors in \(\mathbb{R}^{n}\).

260.1 Examples

example 1 [d] (Linear map \(\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}\) — Dummit and Foote) Let \(V = \mathbb{R}^{3}\) with the standard basis \(B = \{(1,0,0),(0,1,0),(0,0,1)\}\) and let \(W = \mathbb{R}^{2}\) with the standard basis \(E = \{(1,0),(0,1)}\). Let \(\phi\) be the linear transformation

  • \(\phi(x,y,z) = (x + 2y,\, x + y + z)\) .

Since \(\phi(1,0,0) = (1,1)\), \(\phi(0,1,0) = (2,1)\), \(\phi(0,0,1) = (0,1)\), the matrix \(A = M(\phi)\) is

  • \(A = \begin{bmatrix} 1 & 2 & 0 \\ 1 & 1 & 1 \end{bmatrix}\) .

where

  • \(\phi\) is the linear transformation.
  • \(V = \mathbb{R}^{3}\) is the domain.
  • \(W = \mathbb{R}^{2}\) is the codomain.
  • \((x,y,z)\) are coordinates of a vector in \(V\).
  • \(B\) and \(E\) are the standard bases of \(V\) and \(W\).
  • \(A\) is the matrix of \(\phi\) with respect to those bases.

example 2 [d] (Rotation by an angle \(\theta\) — Hubbard and Hubbard) The transformation \(R\) giving rotation by \(\theta\) counterclockwise around the origin is linear, and its matrix is

  • \([R] = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\) .

where

  • \(R: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) is the rotation mapping.
  • \(\theta\) is the angle of rotation.
  • \([R]\) is the matrix of \(R\) with respect to the standard basis of \(\mathbb{R}^{2}\).

260.2 Elementary Example

260.2.1 Simple

A linear transformation maps vectors to vectors and preserves linear combinations.

\[ \phi : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \]

\[ \phi(x,y) = (2x,\ y) \]

\[ \phi(1,0) = (2,0),\quad \phi(0,1) = (0,1) \]

where

  • \(\phi\) is the linear transformation.

260.2.2 General

Rotation by an angle \(\theta\) in the plane is a linear transformation with a \(2 \times 2\) matrix.

\[ R_{\theta} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \]

\[ R_{\theta}(x,y) = (x\cos\theta - y\sin\theta,\ x\sin\theta + y\cos\theta) \]

where

  • \(R_{\theta}\) is the rotation matrix.
  • \(\theta\) is a real angle.

260.3 References

  1. Hubbard, J. H., & Hubbard, B. B. Vector Calculus, Linear Algebra, and Differential Forms, Matrix Editions. — Definition 1.3.2; Example 1.3.9 (rotation by \(\theta\)).
  2. Dummit, D. S., & Foote, R. M. Abstract Algebra. Wiley, 2004. — linear map \(\phi:\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}\), \(\phi(x,y,z)=(x+2y,\,x+y+z)\).