A linear map (or linear transformation) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. In simpler terms, if you have a function $T: V \rightarrow W$ (where V and W are vector spaces), then T is linear if for any vectors $\mathbf{u}, \mathbf{v} \in V$ and any scalar $c$, the following two conditions hold: $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ and $T(c\mathbf{u}) = cT(\mathbf{u})$. Verifying these conditions is key to determining if a function is a linear map.
Match the term with its correct definition:
| Term | Definition |
|---|---|
| 1. Vector Space | A. A function that preserves vector addition and scalar multiplication. |
| 2. Scalar Multiplication | B. A set with operations of addition and scalar multiplication that satisfy certain axioms. |
| 3. Linear Map | C. Multiplying a vector by a scalar. |
| 4. Additivity | D. $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ |
| 5. Homogeneity | E. $T(c\mathbf{u}) = cT(\mathbf{u})$ |
A function $T: V \rightarrow W$ is a ______ ______ if for all vectors $\mathbf{u}, \mathbf{v} \in V$ and any scalar $c$, we have $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ and $T(c\mathbf{u}) = ______$. This means that T preserves vector ______ and ______ multiplication.
Explain, in your own words, why it is important to verify both additivity and homogeneity when determining if a transformation is linear. Can one condition be true while the other is false? Give an example.
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