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Mathematics for Physics

259 Linear Map

A mapping between vector spaces that preserves addition of vectors and multiplication by scalars that is used to transform vectors from one space to another.

definition (Linear Map) A map between vector spaces, \(F: \mathbf{V} \rightarrow \mathbf{W}\), is a linear map if the following conditions hold.

  • additivity \(F(\mathbf{u}+\mathbf{v}) = F(\mathbf{u}) + F(\mathbf{v})\)   for all \(\mathbf{u},\mathbf{v} \in \mathbf{V}\)
  • homogeneity \(F(c\mathbf{v}) = cF(\mathbf{v})\)   for all \(c \in \mathbb{R}, \mathbf{v} \in \mathbf{V}\)

where

  • \(\mathbf{V}, \mathbf{W} \in \mathbb{R}^{n}\) are vector spaces.

259.1 Examples

example 1 [d] (Linear Map \(\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}\) — Dummit and Foote) The map \(\phi : \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}\) defined by

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

is linear. With respect to the standard bases its matrix is

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

where

  • \(\phi\) is the linear map.
  • \((x, y, z)\) are coordinates on the domain \(\mathbb{R}^{3}\).
  • \(A\) is the matrix of \(\phi\) relative to the standard bases.

Note:

  • \(\phi\) maps a vector in \(\mathbb{R}^{3}\) to a vector in \(\mathbb{R}^{2}\).
  • matrix multiplication by \(A\) implements \(\phi\) on column vectors.

259.2 Elementary Example

259.2.1 Simple

A linear map preserves linear combinations. Here a map \(\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\).

\[ T(x_{1},x_{2}) = (x_{1}+x_{2},\ 2 x_{2}) \]

\[ T(e_{1}) = (1,0),\quad T(e_{2}) = (1,2) \]

where

  • \(T\) is the linear map.
  • \(e_{1}, e_{2}\) are the standard basis vectors of \(\mathbb{R}^{2}\).

259.2.2 General

A linear map \(\mathbb{R}^{3} \rightarrow \mathbb{R}^{2}\) is given by a \(2 \times 3\) matrix.

\[ T(\mathbf{x}) = A \mathbf{x},\quad A = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & -1 \end{pmatrix} \]

\[ T(1,0,0) = (1,0),\quad T(0,1,0) = (0,1),\quad T(0,0,1) = (2,-1) \]

where

  • \(A\) is the matrix of \(T\) in the standard bases.