I'm have problem proving: Law for Scalar Multiplication :
Vector spaces possess a collection of specific characteristics and properties. Use the definitions in the attached “Definitions” to complete this task.
Define the elements belonging to $\mathbb{R}^2$ as $\{(a, b) | a, b \in \mathbb{ R}\}$. Combining elements within this set under the operations of vector addition and scalar multiplication should use the following notation:
Vector Addition Example: $(–2, 10) + (–5, 0) = (–2 – 5, 10 + 0) = (–7, 10)$
Scalar Multiplication Example: $–10 × (1, –7) = (–10 × 1, –10 × –7) = (–10, 70)$, where –10 is a scalar.
Under these definitions for the operations, it can be rigorously proven that R2 is a vector space.
Prove Closure under Scalar Multiplication – **i need help with this law **
Can someone put it in a proof form?
Best Answer
If you're asking if vector spaces are closed under multiplication by a scalar, then yes, it is true. If you're asking why, it's because it's written in the definition of a vector space that it must be true ; there is nothing to prove here. It's true because we assume it is when we speak of a vector space.
EDIT : So if I understand this correctly, you need to show that $\mathbb R^2$ is a vector space and you need help showing that $\mathbb R^2$ is closed under scalar multiplication. Scalar multiplication is defined for $\lambda \in \mathbb R$ and $(a,b) \in \mathbb R^2$ via $$ \lambda \cdot (a,b) \overset{def}= (\lambda a, \lambda b) $$ where $\lambda a$ is the usual multiplication of real numbers. What you want to show is that $$ \forall \lambda,a,b \in \mathbb R, \quad \lambda \cdot (a,b) \in \mathbb R^2. $$ Is it obvious now?
Hope that helps,