 So let's see if we can define some vector arithmetic. Since we've defined a vector as an n-tuple whose components come from a field f, we can define the equality of vectors the same way we defined the equality of n-tuples. The corresponding components must be equal. But because the components came from a field, then both addition and multiplication are defined for these components, and so we can define arithmetic operations as well. We'll define these operations component-wise. Given two vectors v with components v1 through vn and u with components u1 through un, and some scalar a, we define a times v, the scalar multiple of v, to be the vector where every component of v is multiplied by a. We'll define v plus u, the sum of the two vectors v at u, to be the vectors whose components are the sum of the corresponding components of the vectors v and u. And finally, we'll define v minus u, the difference between the two vectors, to be the vector v plus negative one times the vector u. Because we can add scalar multiples of vectors, and because anything we do once we can do as many times as we'd like, we can form a linear combination of vectors, a sum of scalar multiples of vectors. So let's take a look at a few examples of scalar multiplication and vector addition. So we'll have two vectors u equal three, one negative two five, and v two one one negative five. Given these vectors, we'll find u plus v, three u, and u minus v. So remember when we add two vectors, we're going to add them component-wise. The first component to the first component, the second to the second, and so on. So we can add these two vectors. We'll start by adding the first components together. Three plus two is five. We'll add the second components together. One plus one is two. We'll add the third components. Negative two plus one is negative one. And finally, adding the fourth components together. Five plus negative five is zero. Which gives us the sum of the vector u plus v, which will be the vector five, two, negative one, zero. What about the vector three u? This is a scalar multiple, so checking our definition, we see that we want to multiply every component of our vector by the given scalar. So the vector three u is going to be found by multiplying every component of the vector u by three. So we'll multiply the first component by three. That gives us nine. We'll multiply the second component by three. That gives us three. We'll multiply the third component by three. That gives us negative six. And we'll multiply the last component by three. That gives us fifteen. And so our vector three u is going to be nine, three, negative six, fifteen. Finally, our vector u minus v. By definition, the difference between two vectors is the first vector plus negative one times the second vector. So that means if I want to find u minus v, what I need to find is u plus negative one times v. So let's find negative one times v. And that's going to be the vector v with every component multiplied by negative one. So that gives us the vector negative two, negative one, negative one, five. Now we'll add our vectors component-wise. The first component of u with the first component of negative v. The second component of u with the second component of negative v. The third components. And the fourth components. Performing the additions gives us our final vector u minus v one, zero, negative three, ten.