What is an identity vector?

So, What Exactly Is an Identity Vector?

Alright, let’s talk identity vectors. Now, you might not hear that term thrown around as much as “identity matrix” or even “zero vector,” but trust me, understanding the idea behind it is super important if you’re diving into linear algebra. Essentially, what we mean by “identity vector” really depends on the situation. It could be the zero vector when we’re talking about adding vectors together, or it could be those special “unit vectors” that make up the columns of an identity matrix. Let’s break it down.

First up: vector addition. Imagine you’re adding something to a vector and you want the original vector to stay exactly the same. What do you add? Zero, right? Well, the zero vector is exactly that – the identity vector for addition. We usually write it as 0→. So, if you’ve got any vector at all, let’s call it v→, then:

v→ + 0→ = v→

Simple as that! The zero vector has the same number of dimensions as your original vector, but every single component is zero. Think of it like this: in a flat, 2D world, it’s (0, 0). In a 3D world? It’s (0, 0, 0). Makes sense, right?

Now, let’s switch gears and talk about identity matrices. Remember those? They’re the square matrices with a diagonal line of ones and zeros everywhere else. Well, the columns of an identity matrix are made up of unit vectors. And sometimes, we can think of these as identity vectors too, especially when we’re talking about how matrices transform things. An identity matrix is like a mirror for transformations. It doesn’t change anything!

The cool thing is, each column in that identity matrix is a unit vector, like e_i. This vector has a ‘1’ in the i-th spot and zeros everywhere else. For example, if we’re in 3D, a 3×3 identity matrix looks like this:

basic