Right now I am struggling as to how to approach this question my lecturer was very poor at conveying this topic.
So here's the question that I am facing with:
If V is any vector space and $ \mathbb c $ is scalar, let $ \mathbb T:V\rightarrow V$ be the function defined by $ \mathbb T(v) =cv$.
a)Show that T is a linear operator(it is called the scalar transformation by $ \mathbb c $).
b)For $V =\mathbb R^2$ sketch $ \mathbb T(1,0)$ and $\mathbb T(0,1)$ in the following cases: (i) $ \mathbb c=2$; (ii) $ \mathbb c=\frac{1}{2}$; (iii) $ \mathbb c=-1$;
1 Answer
$\begingroup$We have to show that $T(\lambda v+\mu w)=\lambda T(v)+\mu T(w)$ for all $v,w\in V$ and $\lambda,\mu\in \mathbb{F}$. Here $\mathbb{F}$ is the base field. In most cases one considers $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. Now by defintion there is some $c\in \mathbb{F}$ such that $T(v)=cv$ for all $v\in V$. Hence $$T(\lambda v +\mu w)=c(\lambda v +\mu w)=\lambda(cv)+\mu(cw)=\lambda T(v)+\lambda T(w).$$ This shows that $T$ is linear. Notice that a linear map can be described as a map that preserves linear combinations of vectors. If you understand this last sentence, you will never forget what a linear map is.
Now do $b)$ yourself.
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