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:
- Additivity: \(T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})\) for all \(\mathbf{u}, \mathbf{v} \in V\)
- 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: \(T((x_1,y_1)+(x_2,y_2)) = T(x_1+x_2, y_1+y_2) = (2(x_1+x_2), (x_1+x_2)+(y_1+y_2)) = (2x_1, x_1+y_1) + (2x_2, x_2+y_2) = T(x_1,y_1) + T(x_2,y_2)\).
Children
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