Find $n + 1$ vectors in $\mathbb{C}^n$ that are linearly independent over $\mathbb{R}$

Question

Consider the vector space $\mathbb{R}^n$ over $\mathbb{R}$ with usual operations.

Find $n + 1$ vectors in $\mathbb{C}^n$ that are linearly independent over $\mathbb{R}$

My attempts: I know that every real number is a complex number... so we can not finds $n + 1$ vectors in $\mathbb{C}^n$ that are linearly independent over $\mathbb{R}$ Because if the $n$ vectors are linearly independent then they are the full set of a basis and the remaining one will surely be a spanned by the other $n$ elements.

Is it True/false?

Any Hints/solution will be appreaciated

thanks in advance

Answer 1

Take the standard basis $\{e_1,\ldots,e_n\}$ of $\mathbb{C}^n$ and add $f=(i,0,0,\ldots,0)$ to it. If $\alpha_1e_1+\cdots\alpha_n e_n+\alpha_{n+1}f=0$, with all $\alpha_k$'s real, then$$\left\{\begin{array}{l}\alpha_1+\alpha_{n+1}i=0\\\alpha_2=0\\\alpha_3=0\\\vdots\\\alpha_n=0\end{array}\right.$$Therefore, all $\alpha_k$'s are equal to $0$:

Answer 2

Note that you are looking for n+1 vectors in $\mathbb{C}^n$ not in $\mathbb{R}^n$

Since $\mathbb{C}^n$ is a $2n$ dimensional space over $\mathbb{R}$, there are many such vectors in $\mathbb{C}^n$

Answer 3

You can remember that $\mathbb{C}^{n}$ is isomorphic to $\mathbb{R}^{2n}$ so you can definitely find n+1 $\mathbb{R}$ independent vectors in $\mathbb{C}^{n}$