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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?

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2 Answers

up vote 3 down vote accepted

To comment on your attempt:

  1. The number of vectors in $M$ can be literally anything from one to infinity. If $M$ was linearly independent, then it would have at most $n$ vectors.

  2. If a number of vectors in a set is smaller than the dimension of the space, you can conclude nothing about the (in)dependence of those vectors.

    If it's strictly greater than $\dim X$ (which is an assumption that you do not have), then you can conclude that its vectors are linearly dependent.

Sasha already gave you an example which shows that your initial statement is not true, so I'll just add that these two are:

If $M$ is a linearly independent set in a complex vector space $X$, then $M$ is also linearly independent in $X$ regarded as a real vector space.

and

If $M$ is a linearly dependent set in a real vector space $X$, then $M$ is also linearly dependent in $X$ regarded as a complex vector space.

These both boil down to the fact that you have a wider choice of factors in $\mathbb{C}$ than you have in $\mathbb{R}$. So, you can have $i \cdot 1 = i$, but not $x \cdot [1\ 0]^T = [0\ 1]^T$, no matter what $x \in \mathbb{R}$ you choose.

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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 $\mathbb{R}$.

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