Question

Show that the vectors (1-i,i) and (2,-I+i) in R² are Linearly Dependent over Field R but Linearly Independent over R , where i = √-1

Answer 1

Answer on Question #43817 – Math – Linear Algebra

Question:

Show that the vectors $(1-i,i)$ and $(2,-1+i)$ in $C^2$ are Linearly Dependent over Field C but Linearly Independent over R, where $i = \sqrt{-1}$

Solution.

Two vectors $v_{1}, v_{2}$ are Linearly Dependent over field $F$ if there exists scalar $a$ in $F$ such that

$$v_{1} = a v_{2}$$

In this case we have $v_{2} = (1 - i, i)$, $v_{1} = (2, -1 + i)$. It can be easily seen that $a = (1 + i)$.

Indeed $av = (1 + i)(1 - i, i) = (2, -1 + i) = v_{1}$.

Hence, this vector is linearly dependent over field $C$. But they are linearly independent over $R$, because $a = (1 + i)$ doesn't belong to $R$.

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