Linear transformations

Definition: A linear transformation (or linear map) is a function (T: V \to W) between vector spaces over the same field that preserves vector addition and scalar multiplication:

1. Additivity: (T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})) for all (\mathbf{u}, \mathbf{v} \in V)
2. Homogeneity: (T(c\mathbf{v}) = cT(\mathbf{v})) for all (\mathbf{v} \in V) and scalars (c)

These two conditions can be combined into a single property:
[T(c_1\mathbf{v}_1 + c_2\mathbf{v}_2) = c_1 T(\mathbf{v}_1) + c_2 T(\mathbf{v}_2)]

Key consequences:

• (T(\mathbf{0}) = \mathbf{0}) (the zero vector always maps to the zero vector)
• (T) is completely determined by its action on a basis of (V)

Example: The map (T: \mathbb{R}^2 \to \mathbb{R}^2) defined by (T(x,y) = (2x, x+y)) is linear. Check: (T((x_1,y_1)+(x_2,y_2)) = T(x_1+x_2, y_1+y_2) = (2(x_1+x_2), (x_1+x_2)+(y_1+y_2)) = (2x_1, x_1+y_1) + (2x_2, x_2+y_2) = T(x_1,y_1) + T(x_2,y_2)).