In quantum physics, you need to know how to use linear operators. An operator A is said to be linear if it meets the following condition:

For instance, the expression

is actually a linear operator. In order to understand this, you need to know just a little more about what happens when you take the products of bras and kets. Firstly, if you take the product of the bra,

where c is a complex number, then you get the answer,

Secondly, if you take the product of the bra,

Now that you know this you can test to see if

is actually a linear operator. Okay then, you can now apply

to a linear combination of kets, like so,

where c1 and c2 are complex numbers. Now that you know how the product of a bra with a sum of two kets goes, you can say,

Then, as you know,

you can finally write this as,

This is exactly what a linear operator should do — if you replace A in the above equation defining a linear operator, with

then the result is the same as the one you just found. So

is indeed a linear operator — although a pretty funny looking one!