Linear Algebra Done Right · Problem 15

This is about verifying that scalar multiplication distributes over addition of scalars—a foundational property of vector spaces. The key is to work component-by-component: expand what $(a+b)x$ means using the definition of scalar multiplication, then show it equals the sum $ax + bx$ by comparing each coordinate. You're not discovering something deep, but rather confirming that the definition of scalar multiplication respects the field structure of $\mathbf{F}$.

This is about verifying that scalar multiplication distributes over addition of scalars—a foundational property of vector spaces. The key is to work component-by-component: expand what $(a+b)x$ means using the definition of scalar multiplication, then show it equals the sum $ax + bx$ by comparing each coordinate. You're not discovering something deep, but rather confirming that the definition of scalar multiplication respects the field structure of $\mathbf{F}$.