I have the following problem from section 1.4 (Vector Spaces) of Peter Peterson's Linear Algebra textbook. I am having trouble with the way multiplication is defined on the given vector space, $\bf{V}^{*}$.
6.Let $\mathbf{V}$ be a complex vector space i.e., a vector space were the scalars are $\mathbb{C}$.
Define $\mathbf{V}^{*}$ as the complex vector space whose additive structure is that of $\mathbf{V}$ but where complex scalar multiplication is given by
$$\lambda\cdot x=\overline{\lambda}x.$$
Show that $\mathbf{V}^{*}$ is a complex vector space.
Does the bar over $\lambda$ on the right-hand side mean anything? Does it have something to do with complex numbers?
Thank you, and I apologize for the trivial question. From the definition, I should be able to easily show that it is a vector space.
Best Answer
It denotes complex conjugation (Wikipedia link). Given a complex number $\lambda=a+ib$, we define its complex conjugate to be $$\overline{\lambda}:=a-ib$$