Trivial

You should expect to hear the adjective trivial several times in a class about linear algebra.

“This rule applies forever example except the trivial example where…”

We are going to make up the informal example: “when you add a second number to a first number you change the first number to a new number except for the trivial example where the second number is 0.”

It might seem reasonable that adding 0 or multiplying by 1 is trivial since we don’t get a change. However, we hesitate to view Identity Operations as being unimportant – – sometimes they are very important. We promise you will see this when you come to Tensors.

Over time, you will probably notice that a example hides something important, ergo, a trivial example is deceptive.

Let’s watch this deception happen for the story about the generalization of addition and multiplication.

General Notions for Addition and Multiplication

If we add a first vector and a second vector the answer is a vector.

Scalar multiplication of a vector takes a vector and a number and the result is a vector.

We generalize the above specifics as follows:

Addition is a binary operation that takes two elements from a set and returns something that is an element from that set. There is only one set in this story.

Multiplication is a binary operation that takes an element from a first set and an element from a second set and returns an element from the first set.

If we only work with numbers then we won’t notice the something else as being a something else since we will be using the number to scale our number.

Do you see how the example with only numbers is trivial?