A Linear Transformation

Let $\mathbf{T}$ be a transformation from a vector space into the same vector space. That is, for any vector $\mathbf{u}$ , $\mathbf{T}\mathbf{u}$ is also a vector of the same dimensions as $\mathbf{u}$ . Then $\mathbf{T}$ is linear if and only if

$\displaystyle \mathbf{T}(\alpha \mathbf{u} + \beta \mathbf{w}) = \alpha \mathbf{T}\mathbf{u} + \beta \mathbf{T}\mathbf{w}\$ for every real $\displaystyle \ \alpha\$ and $\displaystyle \ \beta$

(2.8.1.1)

[NOTE:

is a linear function of

if and only if $f(x) = \alpha x$ where $\alpha$ is a real number. For example

is not a linear function of

even though it is often referred to as such.]
A linear transformation from a vector space into another vector space is also called a second-order tensor.