Linear Map - Examples

Examples

• The identity map and zero map are linear.
• The map, where c is a constant, is linear.
• For real numbers, the map is not linear.
• For real numbers, the map is not linear (but is an affine transformation, and also a linear function, as defined in analytic geometry.)
• If A is a real m × n matrix, then A defines a linear map from Rn to Rm by sending the column vector x ∈ Rn to the column vector Ax ∈ Rm. Conversely, any linear map between finite-dimensional vector spaces can be represented in this manner; see the following section.
• The (definite) integral is a linear map from the space of all real-valued integrable functions on some interval to R
• The (indefinite) integral (or antiderivative) is not considered a linear transformation, as the use of a constant of integration results in an infinite number of outputs per input.
• Differentiation is a linear map from the space of all differentiable functions to the space of all functions.
• If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f : V → W to dim_F(W) × dim_F(V) matrices in the way described in the sequel are themselves linear maps.
• The expected value of a random variable is linear, as for random variables X and Y we have E = E + E and E = aE, but the variance of a random variable is not linear, as it violates the second condition, homogeneity of degree 1: V = a2V.