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**1. The problem statement, all variables and given/known data**

An inner product is linear in both components.

**2. Relevant equations**

<x,y> = <conjugate(y),conjugate(x)>

<x+y,z> = <x,z> +<y,z>

**3. The attempt at a solution**

I thought it was true . It is obvious that it is linear for the first component by definition

Attempt to show it is for second component:

<x,y+z> = <conjugate(y+z),conjugate(x)>

=<conjugate(y),conjugate(x)> + <conjugate(z),conjugate(x)>

= <x,y> + <x,y>

But the answer is false. I am having trouble understanding why it is not linear in both components. The answer key says that the second component is conjugate- linear.