Linearity

What makes a transform linear? This question gets to the heart of linear algebra. The concept of linearity ties together all the other concepts we have considered so far and the ones to come. Indeed, quantum mechanics is a linear theory. That's what makes linear algebra crucial to understanding quantum computing.

Before I define linearity, let's look at what it is not. Real-life examples of non-linearity abound. For example, exercising 1 hour a day for 24 days does not give the same result as exercising 24 hours in 1 day. Watering a plant is another good non-linear example. Giving a plant 1 gallon of water a day for 100 days will be much better than giving it 100 gallons in 1 day. These are both examples of non-linear relationships. How much you put in does not always translate to what you get out.

Linear relationships, on the other hand, are proportional. Speed is a good example. If you go 20 mph for 1 hour, you'll cover 20 miles. If you go 1 mph for...