Having extended the euclidean spaces to general vector spaces, we may also extend linear transformations.

A transformation ** T**:

` T`(

Since linear transformations between euclidean spaces are equivalent to matrices,
the extension can be thought of as a generalization of matrices. Moreover, by taking `c` = 0 in ` T`(

` T`(

By combining the two properties, ** T** also preserves linear combinations

` T`(

Example For any vector space ** V**, the identity transformation

** id**(

is linear. More generally, any scaling map, such as ** u** → 2

For any vector spaces ** V** and

** O**(

sending any vector in ** V** to the zero vector in

** D**(

of taking derivatives is linear. The transformation

** M**(

of multiplying the function `t`^{2}+1 is also linear.

The exponential transformation

` P`(

is not linear because ** P**(

** E**(

at `t` = 2 is linear. The multiple evaluations

** E**(

at `t` = 0, 1, 2 is also linear. The transformation

** C**(

of combining sine and cosine is linear. A more general statement about the linearity of ** C** can be found here.

Example The transpose transformation

** T**(

is linear. The transformation

** M**(

of multiplying a fixed `m` by `k` matrix ** A** on the left is also linear.
More generally, for fixed

** S**(

of taking squares is not linear because ** S**(2