Question

For the vectors $u = (a,b), v = (c,d)$ in $C^{2}$ the inner product of $u$ and $v$ is defined by $\langle u,v \rangle = a\bar{c} + b\bar{d}$. If $u = (1+i, i), v = (i, 1-i)$ then $\langle u,v \rangle = $

• A. 0
• B. $(1, -i)$
• C. $(2i, -2i)$
• D. $(1+i, 1-i)$

Hint:

Don't forget the bar ($\bar{c}$)! In complex spaces, the second vector's components must be conjugated.

Answer 1

The Correct Option is A

Step 1: Concept
The complex inner product involves taking the conjugate of the second vector's components.

Step 2: Meaning
$u = (1+i, i)$ and $v = (i, 1-i)$. The conjugates of $v$ components are $\bar{c} = -i$ and $\bar{d} = 1+i$.

Step 3: Analysis
$\langle u,v \rangle = (1+i)(-i) + (i)(1+i) = (-i - i^2) + (i + i^2)$.
Since $i^2 = -1$, this is $(-i + 1) + (i - 1)$.

Step 4: Conclusion
Summing the terms gives $1 - 1 - i + i = 0$. Final Answer: (A)