Suppose $M$ is a linearly dependent set in a complex vector space $X$, Is $M$ linearly dependent in $X$, regarded as a real vector space??
My attempt
Say $dim X = n$ regarded a comlex vector space. We know number of vectors in $M$ is $<$ than $n$. since $M$ is linearly dependent. We know $\dim X = 2n$ if $X$ is regarded a real vector space. Therefore, number of vectors in $M < n < 2n \implies $ $M$ is linearly dependent if $X$ is regarded as a real vector space. IS this correct?
Best Answer
I don't think it is the case. Elements $1,i\in\mathbb{C}$ ($1$-dimensional complex vector space) are dependent over $\mathbb{C}$, but are independent over $\mathbb{R}$.