What is the bar symbol over a complex scalar in the expression $\overline{\lambda}$?

Question

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.

Answer 1

Recall that $\lambda$ is a scalar, hence a real, or in the case here, a complex number, so it makes sense to talk about $\overline{\lambda}$, the complex conjugate. As a sidenote, do note that if $\bf V$ is a real vector space, then ${\bf V}^\ast = {\bf V}$.

Answer 2

It denotes complex conjugation (Wikipedia link). Given a complex number $\lambda=a+ib$, we define its complex conjugate to be $$\overline{\lambda}:=a-ib$$