# Is linear dependence dependent on field?

Given the vector space $$\mathbb{C}^3$$ and three vectors $$x_1 = (1, 0, 0)$$, $$x_2 = (0, 1, 0)$$, $$x_3 = (0, 0, 1)$$.

Is it correct that these three vectors are linearly independent if $$\mathbb{C}^3$$ is defined over the field $$\mathbb{R}$$, while they are linearly dependent if the field is $$\mathbb{C}$$?

I'm using the following definiton for linear dependence (Halmos, Finite-Dimensional Vector Spaces, 2e):

A finite set of vectors $$\{x_i\}$$ is linearly dependent if there exists a corresponding set $$\{\alpha_i\}$$ of scalars, not all zero, such that

$$\tag{1}\label{eqn_li}\sum_i \alpha_i x_i = 0,$$

and the reason I'm asking is that $$\eqref{eqn_li}$$ is satisfied for the scalars $$\alpha_1 = i$$ , $$\alpha_2 = 0$$, $$\alpha_3 = 0$$, hence there is a set of scalars $$\{\alpha_i\}$$, not all zeros, such that $$\eqref{eqn_li}$$ holds.

• "...for the scalars..." . What scalars?? – DonAntonio Jan 12 at 17:44
• The given set of scalars doesn't satisfy (1). – Thomas Shelby Jan 12 at 17:45
• Hint: $\sum\alpha_jx_j=(\alpha_1,\alpha_2,\alpha_3)$. – David C. Ullrich Jan 12 at 18:03

Those three vectors are linearly independent both over $$\mathbb C$$ and over $$\mathbb R$$.
However, $$(1,0,0)$$ and $$(i,0,0)$$ are linearly dependent over $$\mathbb C$$ and linearly independent over $$\mathbb R$$.
Your assumption that $$ix_1+0x_2+0x_3=0$$ is wrong. Your three vectors are linearly independent, no matter if we view $$\Bbb C^3$$ as three-dimensional space over $$\Bbb C$$, or six-dimensional space over $$\Bbb R$$ (or e.g., infinite-dimensional space over $$\Bbb Q$$) in the apparent way.
However, $$(1,0,0)$$ and $$(i,0,0)$$ are $$\Bbb R$$-linearly independant and $$\Bbb C$$-linearly dependant.