258 Linear Function
A mapping that preserves addition of vectors and multiplication by scalars that is used to express linear relations between vectors.
definition (Linear Function) A function from a vector space, \(f: \mathbf{V} \rightarrow \mathbb{R}\), that satisfies the following conditions for all vectors and scalars:
- (Additivity) \(f(\mathbf{u} + \mathbf{v}) = f(\mathbf{u}) + f(\mathbf{v})\) .
- (Homogeneity) \(f(k\mathbf{v}) = k \cdot f(\mathbf{v})\).
where
- \(\mathbf{V}\) is a real vector space.
- \(\mathbf{u}, \mathbf{v} \in \mathbf{V}\).
- \(k\) is a scalar.
258.1 Elementary Example
258.1.1 Simple
A linear function preserves addition and scalar multiplication.
\[ f : \mathbb{R}^{2} \rightarrow \mathbb{R} \]
\[ f(x_{1},x_{2}) = 2 x_{1} - x_{2} \]
\[ f(e_{1}) = 2,\quad f(e_{2}) = -1 \]
where
- \(f\) is the linear function.
- \(e_{1} = (1,0)\) and \(e_{2} = (0,1)\).
258.1.2 General
On \(\mathbb{R}^{3}\), a linear function is given by three coefficients.
\[ f(x_{1},x_{2},x_{3}) = a_{1} x_{1} + a_{2} x_{2} + a_{3} x_{3} \]
\[ (a_{1},a_{2},a_{3}) = (1,-2,4) \]
where
- \(a_{i}\) are fixed scalars.
- Adjoint
- Basis
- Characteristic Polynomial
- Complex Conjugate
- Conjugate Symmetry
- Conjugate Transpose
- Determinant
- Diag
- Dual Space
- Eigendecomposition
- Eigenvalue
- Eigenvector
- Hermitian
- Hermitian Conjugate
- Homogeneity
- Homogeneous
- Inner Product
- Inner Product Space
- Kronecker Delta
- Linear Function
- Linear Map
- Linear Transformation
- Operator
- Orthonormal
- Orthonormal Set of Functions
- Projection Map
- Quadratic
- Standard Basis