Linear independence of the vector in different vector spaces

Question

Prove or disprove

a) In real vector space $\mathbb R$, set $\{1+i,1-i\}$ is linearly independent?

b) In complex vectors space $\mathbb C$, the set $\{1+i,1-i\}$ is linearly independent?

a) scalar $\alpha_1,\alpha_2 \in \mathbb R$ such that $\alpha_1(1+i)+\alpha_2(1-i)=0$

then I get system $\alpha_1+\alpha_2=0$, $\alpha_1-\alpha_2=0$ so $\alpha_1=\alpha_2=0$ so they are linear independence

b) take some scalar $\alpha_1,\alpha_2 \in \mathbb C$ such that $\alpha_1=a_1+ib_1$, $\alpha_2=a_2+ib_2$, $\alpha_1(1+i)+\alpha_2(1-i)=0$, then I get system $a_1-b_1+a_2+b_2=0$, $a_1+b1-a2+b2=0$, so $b_1=a_2$, $a1=-b2$, so $\alpha_1=i\alpha_2$ so they are linear dependent, is this ok?

Answer 1

Yes for point "b)" the vectors are linearly dependent indeed from the condition $\alpha_1=i\alpha_2$ we have for example

$$i\cdot (1+i)+1\cdot (1-i)=i-1+1-i=0$$

Answer 2

For b): Since $\mathbb C$ is a one-dimensional vector space over itself, any subset of the complex numbers containing more than one element is $\mathbb C$-linearly dependent.